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+# ENHANCED CONVOLUTIONAL NEURAL KERNELS
+
+# Anonymous authors
+
+Paper under double-blind review
+
+# ABSTRACT
+
+Recent research shows that for training with $\ell _ { 2 }$ loss, convolutional neural networks (CNNs) whose width (number of channels in convolutional layers) goes to infinity correspond to regression with respect to the CNN Gaussian Process kernel (CNN-GP) (Novak et al., 2019) if only the last layer is trained, and correspond to regression with respect to the Convolutional Neural Tangent Kernel (CNTK) if all layers are trained. An exact algorithm to compute CNTK (Arora et al., 2019) yielded the finding that classification accuracy of CNTK on CIFAR-10 is within $6 { - } 7 \%$ of that of the corresponding CNN architecture (best figure being around $78 \%$ which is interesting performance for a fixed kernel.
+
+Here we show how to significantly enhance the performance of these kernels using two ideas. (1) Modifying the kernel using a new operation called Local Average Pooling (LAP) which preserves efficient computability of the kernel and inherits the spirit of standard data augmentation using pixel shifts. Earlier papers were unable to incorporate naive data augmentation because of the quadratic training cost of kernel regression. This idea is inspired by Global Average Pooling (GAP), which we show for CNN-GP and CNTK is equivalent to full translation data augmentation. (2) Representing the input image using a pre-processing technique proposed by Coates et al. (2011), which uses a single convolutional layer composed of random image patches.
+
+On CIFAR-10, the resulting kernel, CNN-GP with LAP and horizontal flip data augmentation, achieves $8 9 \%$ accuracy, matching the performance of AlexNet (Krizhevsky et al., 2012) , and outperforms the best previous classifier that is not a trained neural network (Mairal, 2016). Similar improvements are obtained for Fashion-MNIST.
+
+# 1 INTRODUCTION
+
+Recent research shows that for training with $\ell _ { 2 }$ loss, convolutional neural networks (CNNs) whose width (number of channels in convolutional layers) goes to infinity, correspond to regression with respect to the CNN Gaussian Process kernel (CNN-GP) if only the last layer is trained (Novak et al., 2019; Garriga-Alonso et al., 2019), and correspond to regression with respect to the Convolutional Neural Tangent Kernel (CNTK) if all layers are trained (Jacot et al., 2018; Allen-Zhu et al., 2018; Du et al., 2019b; Arora et al., 2019). Novak et al. (2019); Garriga-Alonso et al. (2019) also implemented CNN-GP and tested its empirical performance. An efficient exact algorithm was given (Arora et al., 2019) to compute CNTK for CNN architectures, as well as those that include a Global Average Pooling (GAP) layer (defined below). This is a fixed kernel that inherits some benefits of CNNs, including exploitation of locality via convolution, as well as multiple layers of processing. For CIFAR-10, incorporating GAP into the kernel improves classification accuracy by up to $1 0 \%$ compared to pure convolutional CNTK.
+
+While this performance is encouraging for a fixed kernel, the best accuracy is still under $7 8 \%$ , which is disappointing even compared to AlexNet. One hope for improving the accuracy further is to somehow capture modern innovations such as batch normalization, data augmentation, residual layers, etc. in CNTK. The current paper shows how to incorporate simple data augmentation. Specifically, the idea of creating new training images from existing images using pixel translation and flips, while assuming that these operations should not change the label. Since deep learning uses stochastic gradient descent (SGD), it is trivial to do such data augmentation on the fly. However, it’s unclear how to efficiently incorporate data augmentation in kernel regression, since training time is quadratic in the number of training images.
+
+Thus somehow data augmentation has to be incorporated into the computation of the kernel itself. The main observation here is that the above-mentioned algorithm for computing CNTK involves a dynamic programming whose recursion depth is equal to the depth of the corresponding finite CNN. It is possible to impose symmetry constraints at any desired layer during this computation. In this viewpoint, it can be shown that prediction using CNTK/CNN-GP with GAP is equivalent to prediction using CNTK/CNN-GP without GAP but with full translation data augmentation with wraparound at the boundary. The translation invariance property implicitly assumed in data augmentation is exactly equivalent to an imposed symmetry constraint in the computation of the CNTK which in turn is derived from the pooling layer in the CNN. See Section 4 for more details.
+
+Thus GAP corresponds to full translation data augmentation scheme, but in practice such data augmentation creates unrealistic images (cf. Figure 2) and training on them can harm performance. However, the idea of incorporating symmetry in the dynamic programming leads to a variant we call Local Average Pooling (LAP). This implicitly is like data augmentation where image labels are assumed to be invariant to small translation, say by a few pixels. Interestingly, LAP corresponds to a average pooling layer for CNNs, named box filtering (Szeliski, 2010).
+
+Experimentally, we find LAP significantly enhances the performance as discussed below.
+
+• In extensive experiments on CIFAR-10 and Fashion-MNIST, we find LAP consistently improves performance of CNN-GP and CNTK. In particular, we find CNN-GP with LAP achieves $8 1 \%$ on CIFAR-10 dataset, outperforming the best previous kernel predictor by $3 \%$ .   
+• When using the technique proposed by Coates et al. (2011), which uses randomly sampled patches from training data as filters to do pre-processing,2 CNN-GP with LAP and horizontal flip data augmentation achieves $8 9 \%$ accuracy on CIFAR-10, matching the performance of AlexNet (Krizhevsky et al., 2012) and is the strongest classifier that is not a trained neural network.3 We also test performance of CNNs with an extra layer corresponding to LAP and observe that it improves the performance on certain architectures.
+
+# 2 RELATED WORK
+
+Data augmentation has long been known to improve the performance of neural network and kernel methods (Sietsma & Dow, 1991; Scholkopf et al.¨ , 1996). Theoretical study of data augmentation dates back to Chapelle et al. (2001). Recently, Dao et al. (2018) proposed a theoretical framework for understanding data augmentation and showed data augmentation with a kernel classifier can have feature averaging and variance regularization effects. More recently, Chen et al. (2019) quantitatively shows in certain settings, data augmentation provably improves the classifier performance. For more comprehensive discussion on data augmentation and its properties, we refer readers to Dao et al. (2018); Chen et al. (2019) and references therein.
+
+CNN-GP and CNTK correspond to infinitely wide CNN with different training strategies (only training the top layer or training all layers jointly). The correspondence between infinite neural networks and kernel machines was first noted by Neal (1996). More recently, this was extended to deep and convolutional neural networks (Lee et al., 2018; Matthews et al., 2018; Novak et al., 2019; Garriga-Alonso et al., 2019). These kernels correspond to infinitely wide neural networks where only the last layer is trained. A recent line of work studied overparameterized neural networks where all layers are trained (Allen-Zhu et al., 2018; Du et al., 2019b; 2018; Li & Liang, 2018; Zou et al., 2018). Their proofs imply the gradient kernel is close to a fixed kernel which only depends the training data and neural network architecture. These kernels thus correspond to infinitely wide neural networks where are all layers are trained. Jacot et al. (2018) named this kernel, neural tangent kernel (NTK). Arora et al. (2019) formally proved infinitely wide neural net predictor trained by gradient descent is equivalent to NTK predictor. Recently, NTKs induced by various neural network architectures are derived and shown to achieve strong empirical performance (Arora et al., 2019; Yang, 2019; Du et al., 2019a).
+
+Global Average Pooling (GAP) was first proposed in Lin et al. (2013) and is common in modern CNN design (Springenberg et al., 2014; He et al., 2016; Huang et al., 2017). However, current theoretical understanding on GAP is still rather limited. It has been conjectured in Lin et al. (2013) that GAP reduces the number of parameters in the last fully-connected layer and thus avoids overfitting, and that GAP is more robust to spatial translations of the input since it sums out the spatial information. In this work, we study GAP from the CNN-GP and CNTK perspective, and draw an interesting connection between GAP and data augmentation.
+
+Here we are interested in methods that are not trained neural networks. If the features are predefined before seeing the data, Oyallon & Mallat (2015) proposed the scattering network which achieves $\cdot$ classification accuracy on CIFAR-10. If one uses unsupervised learning methods to extract features, the method proposed in Coates et al. (2011) is one of the best-performing approaches on CIFAR-10 preceding modern CNNs. To our knowledge, the best result via unsupervised learning method in this line is by Mairal (2016), who used the convolutional kernel network to achieve $\cdot$ accuracy on CIFAR-10. In this work we combine CNTK with LAP and the idea in Coates et al. (2011) to achieve the best performance for classifiers that are not trained neural networks.
+
+# 3 PRELIMINARIES
+
+# 3.1 NOTATION
+
+We use bold-faced letters for vectors, matrices and tensors. For a vector $\textbf { \em a }$ , let $[ \pmb { a } ] _ { i }$ be its $i$ -th entry; for a matrix $\pmb { A }$ , let $[ A ] _ { i , j }$ be its $( i , j )$ -th entry; for an order 4 tensor $_ { \mathbf { T } }$ , let $[ \bar { \pmb { T } } ] _ { i j , i ^ { \prime } j ^ { \prime } }$ be its $( i , j , i ^ { \prime } , j ^ { \prime } )$ -th entry. For a symmetric tensor, wet let $\begin{array} { r } { \mathrm { t r } \left( \pmb { T } \right) = \sum _ { i , j } \pmb { T } _ { i j , i j } } \end{array}$ . For an order $d$ tensor $\pmb { T } \in \mathbb { R } ^ { C _ { 1 } \times C _ { 2 } \times . . . \times C _ { d } }$ and an integer $\alpha \in [ C _ { d } ]$ , we use $\pmb { T } _ { ( \alpha ) } \in \mathbb { R } ^ { C _ { 1 } \times C _ { 2 } \times . . . \times C _ { d - 1 } }$ to denote the order $d - 1$ tensor formed by fixing the coordinate of the last dimension of $_ { \mathbf { T } }$ to be $\alpha$ .
+
+# 3.2 CNN, CNN-GP AND CNTK
+
+In this section we give formal definitions of CNN, CNN-GP and CNTK that we study in this paper. Throughout the paper, we let $P$ be the width and $Q$ be the height of the image. We use $q \in \mathbb { Z } _ { + }$ to denote the filter size. In practice, $q = 1 , 3 , 5$ or 7.
+
+Padding Schemes. In the definition of CNN, CNTK and CNN-GP, we may use different padding schemes. Let $\pmb { x } \in \mathbb { R } ^ { P \times Q }$ be an matrix. For a given index pair $( i , j )$ with $i \leq 0$ , $i \geq P + 1$ , $j \le 0$ or $j \geq Q + 1$ , different padding schemes define different value for $[ \pmb { x } ] _ { i , j }$ . For circular padding, we define $[ \pmb { x } ] _ { i , j }$ to be $[ { \pmb x } ] _ { i \mathrm { ~ m o d ~ } P , j \mathrm { ~ m o d ~ } G }$ . For zero padding, we simply define $[ \pmb { x } ] _ { i , j }$ to be 0. Note the difference between circular padding and zero padding occurs only on the boundary of images. We will prove our theoretical results for the circular padding scheme to avoid boundary effects.
+
+CNN. Now we describe CNN with and without GAP. For any input image $_ { \textbf { \em x } }$ , after $L$ intermediate layers, we obtain $\pmb { x } ^ { ( L ) } \in \mathbb { R } ^ { P \times Q \times C ^ { ( L ) } }$ where $C ^ { ( L ) }$ is the number of channels of the last layer. See Section A for the definition of $\pmb { x } ^ { ( L ) }$ . For the output, there are two choices: with and without GAP.
+
+• Without GAP: the final output is defined as f(θ, x) = PC(L)α=1 $\begin{array} { r } { f ( \pmb { \theta } , \pmb { x } ) = \sum _ { \alpha = 1 } ^ { C ^ { ( L ) } } \left. \pmb { W } _ { ( \alpha ) } ^ { ( L + 1 ) } , \pmb { x } _ { ( \alpha ) } ^ { ( L ) } \right. } \end{array}$ where $\pmb { x } _ { ( \alpha ) } ^ { ( L ) } \in$ $\mathbb { R } ^ { P \times Q }$ , and ${ W _ { ( \alpha ) } ^ { ( L + 1 ) } } \in \mathbb { R } ^ { P \times Q }$ is the weight of the last fully-connected layer. • With GAP: the final output is defined as f (θ, x) = 1P Q PC α=1 $\begin{array} { r } { f ( \pmb { \theta } , \pmb { x } ) = \frac { 1 } { P Q } \sum _ { \alpha = 1 } ^ { C ^ { ( L ) } } \pmb { W } _ { ( \alpha ) } ^ { ( L + 1 ) } { \cdot } \sum _ { ( i , j ) \in [ P ] \times [ Q ] } \left[ \pmb { x } _ { ( \alpha ) } ^ { ( L ) } \right] _ { i , j } } \end{array}$ where W (L+1) ${ W _ { ( \alpha ) } ^ { ( L + 1 ) } \in \mathbb { R } }$ is the weight of the last fully-connected layer.
+
+CNN-GP and CNTK. Now we describe CNN-GP and CNTK. Let ${ \boldsymbol { x } } , { \boldsymbol { x } } ^ { \prime }$ be two input images. We denote the $L$ -th layer’s CNN-GP kernel as $\Sigma ^ { ( L ) } \left( \pmb { x } , \pmb { x } ^ { \prime } \right) \in \mathbb { R } ^ { [ P ] \times [ Q ] \times [ P ] \times [ Q ] }$ and the $L$ -th layer’s CNTK kernel as ${ \bf \dot { \Theta } } ^ { ( L ) } \left( { \bf x } , { \bf x } ^ { \prime } \right) \in \mathbb { R } ^ { \left[ P \right] \times \left[ Q \right] \times \left[ P \right] \times \left[ \dot { Q } \right] }$ . See Section A for the precise definitions of $\pmb { \Sigma } ^ { ( L ) } \left( \pmb { x } , \pmb { x } ^ { \prime } \right)$ and $\Theta ^ { ( L ) } \left( \pmb { x } , \pmb { x } ^ { \prime } \right)$ . For the output kernel value, again, there are two choices, without GAP (equivalent to using a fully-connected layer) or with GAP.
+
+• Without GAP: the output of CNN-GP is $\Sigma _ { \mathsf { F C } } \left( \mathbf { { x } } , \mathbf { { x } } ^ { \prime } \right) = \mathrm { { \ t r } } \left( \Sigma ^ { ( L ) } ( \mathbf { { x } } , \mathbf { { x } } ^ { \prime } ) \right)$ and the output of CNTK is $\Theta _ { \mathsf { F C } } \left( \pmb { x } , \pmb { x } ^ { \prime } \right) = \mathrm { t r } \left( \Theta ^ { ( L ) } ( \pmb { x } , \pmb { x } ^ { \prime } ) \right)$ . • With $\begin{array} { r l } & { \mathrm { W i t h } \qquad \mathsf { G A P : } \qquad \mathrm { t h e } \qquad \mathrm { o u t p u t } \qquad \mathrm { o f } \qquad \mathsf { C N N - G P } \qquad \mathrm { i s } \qquad \Sigma _ { \mathsf { G A P } } \left( x , x ^ { \prime } \right) } \\ & { \frac { 1 } { P ^ { 2 } Q ^ { 2 } } \sum _ { i , j , i ^ { \prime } , j ^ { \prime } \in [ P ] \times [ Q ] \times [ Q ] } \left[ \mathbf { \Sigma } \mathbf { { C } } ^ { ( L ) } \left( x , x ^ { \prime } \right) \right] _ { i , j , i ^ { \prime } , j ^ { \prime } } , \quad \mathrm { a n d } \quad \mathrm { t h e } \qquad \mathrm { o u t p u t } \quad \mathrm { o f } \quad \mathsf { C N T K } \quad \mathrm { i s } } \\ & { \Theta _ { \mathsf { G A P } } \left( x , x ^ { \prime } \right) = \frac { 1 } { P ^ { 2 } Q ^ { 2 } } \sum _ { i , j ^ { \prime } , i ^ { \prime } , j ^ { \prime } \in [ P ] \times [ Q ] \times [ P ] \times [ Q ] } \left[ \mathbf { \Sigma } \mathbf { { C } } ^ { ( L ) } \left( x , x ^ { \prime } \right) \right] _ { i , j , i ^ { \prime } , j ^ { \prime } } . } \end{array}$
+
+Kernel Prediction. Lastly, we recall the formula for kernel regression. For simplicity, throughout the paper,with data me all ke, define ble. Giwhere ${ \bf K } \left( { \pmb x } , { \pmb x } ^ { \prime } \right)$ and a dataset . The predic $( X , y )$ $\{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { N }$ ${ \bf K } _ { \bf X } \in \mathbb { R } ^ { N \times N }$ $[ \mathbf { K } _ { \mathbf { X } } ] _ { i , j } = \mathbf { K } ( \pmb { x } _ { i } , \pmb { x } _ { j } )$ unseen data $\mathbf { x } ^ { \prime }$ is $\begin{array} { r } { \sum _ { i = 1 } ^ { N } \alpha _ { i } \mathbf { K } ( \pmb { x } ^ { \prime } , \pmb { x } _ { i } ) } \end{array}$ , where $\pmb { \alpha } = \mathbf { K } _ { \mathbf { X } } ^ { - 1 } \pmb { y }$ .
+
+# 3.3 DATA AUGMENTATION SCHEMES
+
+In this paper we consider two types of data augmentation schemes: translation and horizontal flip.
+
+Translation. Given $( i , j ) \in [ P ] \times [ Q ]$ , we define the translation operator $\mathcal { T } _ { i j } : \mathbb { R } ^ { P \times Q \times C } $ $\mathbb { R } ^ { P \times Q \times C }$ : for an image $\pmb { x } \in \mathbb { R } ^ { P \times Q \times C }$ , $\left[ \mathcal { T } _ { i j } \left( \pmb { x } \right) \right] _ { i ^ { \prime } , j ^ { \prime } , c } = \left[ \pmb { x } \right] _ { i ^ { \prime } + i , j ^ { \prime } + j , c }$ for $( i ^ { \prime } , j ^ { \prime } , c ) \in [ P ] \times [ Q ] \times$ $[ C ]$ . Here the precise definition of $[ { \pmb x } ] _ { i ^ { \prime } + i , j ^ { \prime } + j , c }$ depends on the padding scheme. Given a dataset $D = \{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { N }$ , the full translation data augmentation scheme creates a new dataset $D _ { \mathcal { T } } =$ $\{ ( \mathcal T _ { i j } ( x _ { i } ) , y _ { i } ) \} _ { ( i , j , n ) \in [ P ] \times [ Q ] \times [ N ] }$ and training is performed on $D _ { \mathcal { T } }$ .
+
+eratfor $\mathcal { F } : \mathbb { R } ^ { P \times Q \times C }  \mathbb { R } ^ { P \times Q \times C }$ : for an imen a dataset $\pmb { x } ~ \in ~ \mathbb { R } ^ { P \times Q \times C }$ $[ \mathcal { F } ( \pmb { x } ) ] _ { i , j , c } = [ \pmb { x } ] _ { P + 1 - i , j , c }$ $( i , j , c ) \in [ P ] \times [ Q ] \times [ C ]$ $D = \{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { N }$ the horizontal flip augmentation scheme creates a new dataset of the form $D _ { \mathcal { F } } = \{ \left( \mathcal { F } \left( \mathbf { x } _ { i } \right) , y _ { i } \right) \} _ { i = 1 } ^ { N }$ and training is performed on $D _ { \mathcal { F } } \cup D$ .
+
+# 4 EQUIVALENCE BETWEEN AUGMENTED KERNEL AND DATA AUGMENTATION
+
+In this section, we demonstrate the equivalence between using data augmentation and using a augmented kernel. To formally discuss the equivalence, we use group theory to describe translation and horizontal flip operators. We provide the definition of group in Section B for completeness.
+
+It is easy to verify that $\{ \mathcal { F } , \mathcal { Z } \}$ , $\{ \mathcal T _ { i , j } \} _ { ( i , j ) \in [ P ] \times [ Q ] }$ , $\{ \mathcal { T } _ { i , j } \circ \mathcal { F } \} _ { ( i , j ) \in [ P ] \times [ Q ] } \cup \{ \mathcal { T } _ { i , j } \} _ { ( i , j ) \in [ P ] \times [ Q ] }$ are groups, where $\mathcal { T }$ is the identity map. From now on, given a dataset $\left( \mathbf { X } , \pmb { y } \right)$ with data $\{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { N }$ and a group $\mathcal { G }$ , the augmented dataset $\left( \mathbf { X } _ { \mathcal { G } } , \mathbf { y } _ { \mathcal { G } } \right)$ is defined to be $\{ g ( \pmb { x } _ { i } ) , y _ { i } \} _ { g \in \mathscr { G } , i \in [ N ] }$ . Fo r kernel prediction for unseen data $\mathbf { x } ^ { \prime }$ on the augmented dataset, we have the following formula: $\begin{array} { r } { \sum _ { i \in [ N ] , g \in \mathcal { G } } \widetilde { \alpha } _ { i , g } \mathbf { K } ( \pmb { x } ^ { \prime } , g ( \pmb { x } _ { i } ) ) } \end{array}$ , where $\widetilde { \pmb { \alpha } } = \mathbf { K } _ { \mathbf { X } _ { \mathcal { G } } } ^ { - 1 } \pmb { y } _ { \mathcal { G } }$ .
+
+To proceed, we define the concept of augmented kernel. Let $\mathcal { G }$ be a finite group. Define the augmented kernel $\mathbf { K } ^ { \mathcal { G } }$ as $\mathbf { K } ^ { \mathcal { G } } ( \pmb { x } , \pmb { x } ^ { \prime } ) \overset { \cdot } { = } \mathbb { E } _ { \pmb { g } \in \mathcal { G } } \mathbb { \bar { E } } _ { \pmb { g } ^ { \prime } \in \mathcal { G } } \mathbf { K } ( \pmb { g } ( \pmb { x } ) , \pmb { g } ^ { \prime } ( \pmb { x } ^ { \prime } ) )$ where ${ \boldsymbol { x } } , { \boldsymbol { x } } ^ { \prime }$ are two inputs images. A key observation is that for CNTK and CNN-GP, when circular padding and GAP is adopted, these are actually the augmented kernels with the group $\mathcal { G } = \{ \mathcal { T } _ { i , j } \} _ { ( i , j ) \in [ P ] \times [ Q ] }$ . Formally, we have $\begin{array} { r } { \Sigma _ { \mathsf { G A P } } \left( \pmb { x } , \pmb { x } ^ { \prime } \right) = \frac { 1 } { P Q } \pmb { \Sigma } _ { \mathsf { F C } } ^ { \mathcal { G } } \left( \pmb { x } , \pmb { x } ^ { \prime } \right) } \end{array}$ and $\begin{array} { r } { \Theta _ { \mathsf { G A P } } \left( \pmb { x } , \pmb { x } ^ { \prime } \right) = \frac { 1 } { P Q } \Theta _ { \mathsf { F C } } ^ { \mathcal { G } } \left( \pmb { x } , \pmb { x } ^ { \prime } \right) } \end{array}$ . The proof for these two equations is just by checking the formula of these kernels and using definition of circular padding. By similar proof, one can observe the following invariance property of $\Sigma _ { \mathsf { G A P } } , \Sigma _ { \mathsf { F C } } , \Theta _ { \mathsf { G A P } }$ and $\Theta _ { \mathsf { F C } }$ , under all groups mentioned above, including $\{ \mathcal { F } , \mathcal { Z } \}$ and $\{ \mathcal { T } _ { i , j } \} _ { ( i , j ) \in [ P ] \times [ Q ] }$ .
+
+Definition 4.1. A kernel $\mathbf { K }$ is invariant under a group $\mathcal { G }$ if and only if for any $g \in \mathcal G$ $\mathbf { K } ( g ( \pmb { x } ) , g ( \pmb { x } ^ { \prime } ) ) = \mathbf { K } ( \pmb { x } , \pmb { x } ^ { \prime } )$ .
+
+Now the following theorem formally states the equivalence between using an augmented kernel on the dataset and using the kernel on the augmented dataset.
+
+Theorem 4.1. Given a group $\mathcal { G }$ and a kernel $\mathbf { K }$ such that $\mathbf { K }$ is invariant under $\mathcal { G }$ , then the prediction nted kernel . Namely, f $\mathbf { K } ^ { \mathcal { G } }$ wny $\left( \mathbf { X } , \pmb { y } \right)$ $\mathbf { K }$ $\left( \mathbf { X } _ { \mathcal { G } } , \mathbf { y } _ { \mathcal { G } } \right)$ $\pmb { x } ^ { \prime } \in \mathbb { R } ^ { P \times Q \times C }$ $\begin{array} { r } { \sum _ { i = 1 } ^ { N } \alpha _ { i } \mathbf { K } ^ { \mathcal { G } } ( \pmb { x } ^ { \prime } , \pmb { x } _ { i } ) = \sum _ { i \in [ N ] , g \in \mathcal { G } } \widetilde { \alpha } _ { i , g } \mathbf { K } ( \pmb { x } ^ { \prime } , g ( \pmb { x } _ { i } ) ) } \end{array}$ where $\pmb { \alpha } = \left( \mathbf { K } _ { \mathbf { X } } ^ { \mathcal { G } } \right) ^ { - 1 } \pmb { y } , \tilde { \pmb { \alpha } } = \left( \mathbf { K } _ { \mathbf { X } _ { \mathcal { G } } } \right) ^ { - 1 } \pmb { y } \mathcal { G } .$ .
+
+The proof is deferred to Appendix B. Two corollaries are directly followed.
+
+Corollary 4.1. For $\mathcal { G } = \{ \mathcal { T } _ { i , j } \} _ { ( i , j ) \in [ P ] \times [ Q ] }$ , for any given dataset $D$ , the prediction of $\Sigma _ { G A P }$ (or $\Theta _ { G A P } )$ with dataset $D$ is equal to the prediction of $\Sigma _ { F C } ( o r \Theta _ { F C } )$ with augmented dataset ${ \cal D } \tau$ . Corollary 4.2. For $\mathcal { G } = \{ \bar { \mathcal { F } } , \mathcal { Z } \}$ , for any given dataset $D$ , the prediction of $\Sigma _ { G A P } ^ { \mathcal { G } }$ (or $\Theta _ { G A P } ^ { g } )$ with dataset $D$ is equal to the prediction of $\Sigma _ { G A P }$ (or $\Theta _ { G A P , }$ ) with augmented dataset $D _ { \mathcal { F } } \cup D$ .
+
+Now we discuss implications of Theorem 4.1 and its corollaries. Naively applying data augmentation, with full translation on CNTK or CNN-GP for example, one needs to create a $P ^ { 2 } Q ^ { 2 }$ times larger kernel matrix since there are $P Q$ translation operators, which is often computationally infeasible. Instead, we can directly use the augmented kernel $\scriptstyle \sum _ { \mathsf { G A P } }$ or $\Theta _ { \mathsf { G A P } }$ for the case of full translation on CNTK or CNN-GP) for prediction, for which one only needs to create a kernel matrix that is as large as the original one. For horizontal flip, although the augmentation kernel is not as conveniently computed as full translation, Corollary 4.2 still provides a more efficient method for computing kernel value and solving kernel regression, since the augmented dataset is twice as large as the original dataset.
+
+# 5 LOCAL AVERAGE POOLING
+
+In this section, we introduce a new operation called Local Average Pooling (LAP). As discussed in the introduction, full translation data augmentation can create unrealistic images. A natural idea is to do local translation data augmentation, i.e., restricting the distance of translation. More specifically, we only allow translation operations $\mathcal { T } _ { \Delta _ { i } , \Delta _ { j } }$ (cf. Section 3.3) for $( \Delta _ { i } , \Delta _ { j } ) \in [ - c , c ] \times [ - c , c ]$ where $c$ is a parameter to control the amount of allowed translation. With a proper choice of the parameter $c$ , translation data augmentation will not create unrealistic images (cf. Figure 2). However, naive local translation data augmentation is computationally infeasible for kernel methods, even for moderate choice of $c$ . To remedy this issue, in this section we introduce LAP, which is inspired by the connection between full translation data augmentation and GAP on CNN-GP and CNTK. Here, for simplicity, we assume $P = Q$ and derive the formula only for CNTK. Our formula can be generalized to CNN-GP in a straightforward manner.
+
+Recall that for two given images $_ { \textbf { \em x } }$ and $\mathbf { x } ^ { \prime }$ , without GAP, the formula for output of CNTK is $\operatorname { t r } \left( \Theta ( \pmb { x } , \pmb { x } ^ { \prime } ) \right)$ . With GAP, the formula for output of CNTK is $\begin{array} { r } { \frac { 1 } { P ^ { 4 } } \sum _ { i , j , i ^ { \prime } , j ^ { \prime } \in [ P ] ^ { 4 } } \big [ \Theta \left( \pmb { x } , \pmb { x } ^ { \prime } \right) \big ] _ { i , j , i ^ { \prime } , j ^ { \prime } } } \end{array}$ . With circular padding, the formula can be rewritten as $\begin{array} { r } { \frac { 1 } { P ^ { 2 } } \mathbb { E } _ { \Delta _ { i } , \Delta _ { i } ^ { \prime } , \Delta _ { j } , \Delta _ { j } , \Delta _ { j } ^ { \prime } \sim [ P ] ^ { 4 } } \sum _ { i , j \in [ P ] \times [ P ] } \left[ \Theta \left( \pmb { x } , \pmb { x } ^ { \prime } \right) \right] _ { i + \Delta _ { i } , j + \Delta _ { j } , i + \Delta _ { i } ^ { \prime } , j + \Delta _ { j } ^ { \prime } } } \end{array}$ , which is again equal to $\begin{array} { r } { \frac { 1 } { P ^ { 2 } } \mathbb { E } _ { \Delta _ { i } , \Delta _ { i } ^ { \prime } , \Delta _ { j } , \Delta _ { j } ^ { \prime } \sim [ P ] ^ { 4 } } \mathrm { t r } \Big ( \Theta \left( \mathcal { T } _ { \Delta _ { i } , \Delta _ { j } } ( \pmb { x } ) , \mathcal { T } _ { \Delta _ { i } ^ { \prime } , \Delta _ { j } ^ { \prime } } ( \pmb { x } ^ { \prime } ) \right) \Big ) . \mathrm { \ b w } } \end{array}$ e ignore the $1 / P ^ { 2 }$ scaling factor since it plays no role in kernel regression.
+
+Now we consider restricted translation operations $\mathcal { T } _ { \Delta _ { i } , \Delta _ { j } }$ with $( \Delta _ { i } , \Delta _ { j } ) \in [ - c , c ] \times [ - c , c ]$ and derive the formula for LAP. Assuming circular padding, we have
+
+$$
+\begin{array} { r l } & { \quad \mathbb { E } _ { \Delta _ { i } , \Delta _ { i } ^ { \prime } , \Delta _ { j } , \Delta _ { j } ^ { \prime } \sim [ - c , c ] ^ { 4 } } \mathrm { t r } \left( \Theta \left( { \mathcal T } _ { \Delta _ { i } , \Delta _ { j } } ( x ) , { \mathcal T } _ { \Delta _ { i } ^ { \prime } , \Delta _ { j } ^ { \prime } } ( x ^ { \prime } ) \right) \right) } \\ & { = \frac { 1 } { ( 2 c + 1 ) ^ { 4 } } \sum _ { \Delta _ { i } , \Delta _ { i } ^ { \prime } , \Delta _ { j } , \Delta _ { j } ^ { \prime } \in [ - c , c ] ^ { 4 } } \sum _ { i , j \in [ P ] ^ { 2 } } \left[ \Theta ( x , x ^ { \prime } ) \right] _ { i + \Delta _ { i } , j + \Delta _ { j } , i + \Delta _ { i } ^ { \prime } , j + \Delta _ { j } ^ { \prime } } . } \end{array}
+$$
+
+Now we have derived the formula for LAP which is given in Equation 1. Notice that the formula in Equation 1 is a well-defined quantity for all padding schemes. In particular, assuming zero padding, when $c = P$ , LAP is equivalent to GAP. When $c = 0$ , LAP is equivalent to no pooling layer. Another advantage of LAP is that it does not incur significant additional computational cost, since the formula in Equation 1 can be rewritten as $\begin{array} { r } { \sum _ { i , j , i ^ { \prime } , j ^ { \prime } \in [ P ] ^ { 4 } } [ \pmb { w } ] _ { i , j , i ^ { \prime } , j ^ { \prime } } \cdot \left[ \Theta ( \pmb { x } , \pmb { x } ^ { \prime } ) \right] _ { i , j , i ^ { \prime } , j ^ { \prime } } } \end{array}$ where each entry in the weight tensor $\pmb { w }$ can be calculated in $O ( 1 )$ time.
+
+Note that the GAP operation in CNN-GP and CNTK corresponds to the GAP layer in CNNs. Here we observe the following box filtering layer that corresponds to LAP in CNNs. Box filtering layer (BF) is a function RP ×Q → RP ×Q such that [BF(x)]i,j = 1(2c+1)2 P∆i,∆j∈[−c,c]2 xi+∆i,j+∆j . This is in fact a standard average pooling layer but with stride 1 and pooling size $2 c + 1$ . We prove the equivalence between LAP and box filtering layer in Appendix C. In Section 6.3, we test BF on CNNs to verify its effectiveness.
+
+# 6 EXPERIMENTS
+
+In this section we present our empirical findings on CIFAR-10 (Krizhevsky, 2009) and FashionMNIST (Xiao et al., 2017). The detailed experimental setup is reported in Appendix D. When reporting test accuracies, the best result on the test set is in boldface and the result that corresponds to the hyper-parameter chosen by cross-validation is underlined.
+
+# 6.1 ABLATION STUDY ON CIFAR-10 AND FASHION-MNIST
+
+We perform experiments to study the effect of different values of the $c$ parameter in LAP and horizontal flip data argumentation on CNTK and CNN-GP. For experiments in this section we set the bias term in CNTK and CNN-GP to be $\beta = 0$ (cf. Section A). We use the same architecture for CNTK and CNN-GP as in Arora et al. (2019). I.e., we stack multiple convolutional layers before the final pooling layer. We use $d$ to denote the number of convolutions layers, and in our experiments we set $d$ to be 5, 8, 11 or 14, to study the effect of depth on CNTK and CNN-GP. For CIFAR-10, we set the $c$ parameter in LAP to be $0 , 4 , \ldots , 3 2$ , while for Fashion-MNIST we set the $c$ parameter in LAP to be $0 , 4 , \ldots , 2 8$ . Notice that when $c = 3 2$ for CIFAR-10 or $c = 2 8$ for Fashion-MNIST, LAP is equivalent to GAP, and when $c = 0$ , LAP is equivalent to no pooling layer. Results on CIFAR-10 are reported in Tables 1 and 3. Due to space constraint, results on Fashion-MNIST are reported in Tables 5 and 6 in Appendix E. In each table, for each combination of $c$ and $d$ , the first number is the test accuracy without horizontal flip data augmentation (in percentage), and the second number (in parentheses) is the test accuracy with horizontal flip data augmentation. To perform cross-validation to choose the hyper-parameters, we use the last 10000 samples in the training set of CIFAR-10 and Fashion-MNIST as the validation set and the rest samples as the training set. We then use the full training set to report the test accuracy. To perform cross-validation, we choose $\cdot$ , $d$ , CNN or CNN-GP, and whether or not to adopt horizontal flip based on the validation accuracy (shown in Appendix F). With cross-validation, the resulting accuracy is $8 2 . 0 9 \%$ on CIFAR-10 and $9 4 . 0 7 \%$ on Fashion-MNIST.
+
+We made the following observations regarding our experimental results.
+
+• LAP with a proper choice of the parameter $c$ significantly improves the performance of CNTK and CNN-GP. On CIFAR-10, the best-performing value of $c$ is $c = 1 2$ or 16, while on FashionMNIST the best-performing value of $c$ is $c = 4$ . We suspect this difference is due to the nature of the two datasets: CIFAR-10 contains real-life images and thus allow more translation, while Fashion-MNIST contains images with centered clothes and thus allow less translation. For both datasets, the best-performing value of $c$ is consistent across all settings (depth, CNTK or CNNGP) that we have considered. Horizontal flip data augmentation is less effective on Fashion-MNIST than on CIFAR-10. There are two possible explanations for this phenomenon. First, most images in Fashion-MNIST are nearly horizontally symmetric (e.g., T-shirts and bags). Second, CNTK and CNN-GP have already achieved a relatively high accuracy on Fashion-MNIST, and thus it is reasonable for horizontal flip data augmentation to be less effective on this dataset.   
+• Finally, for CNTK, when $c = 0$ (no pooling layer) and $c = 3 2$ (GAP) our reported test accuracies are close to those in Arora et al. (2019) on CIFAR-10. For CNN-GP, when $c = 0$ (no pooling layer) our reported test accuracies are close to those in Novak et al. (2019) on CIFAR-10 and Fashion-MNIST. This suggests that we have reproduced previous reported results.
+
+# 6.2 IMPROVING PERFORMANCE ON CIFAR-10 USING RANDOM PATCHES LAYER
+
+Finally, we explore another interesting question: what is the best performance achievable via a method that is not a trained neural network? To further improve the performance, we combine CNTK and CNN-GP with LAP, together with the unsupervised learning approach developed in Coates et al. (2011). Here we use the variant implemented in Recht et al. (2019). More specifically, we first sample 2048 random image patches with size $5 \times 5$ from all training images. Then for the sampled images patches, we subtract the mean of the patches, then normalize them to have unit norm, and finally perform ZCA transformation to the resulting patches. We use the resulting patches as 2048 filters of a convolutional layer with kernel size 5, stride 1 and no dilation or padding. For an input image $_ { \textbf { \em x } }$ , we use $\mathtt { c o n v } ( { \pmb x } )$ to denote the output of the convolutional layer. As in the implementation in Recht et al. (2019), we use ReLU( $\mathsf { c o n v } ( \pmb { x } ) - \beta _ { \mathrm { f e a t u r e } } )$ and $\mathrm { R e L } \dot { \mathrm { U } } ( - \mathrm { c o n v } ( { \pmb x } ) - \bar { \beta } _ { \mathrm { f e a t u r e } } )$ as the input feature for CNTK and CNN-GP. Here we fix $\beta _ { \mathrm { f e a t u r e } } = 1$ as in Recht et al. (2019) and the bias term $\beta$ in CNTK and CNN-GP to be $\beta = 1$ . To make the output kernel value invariant under horizontal flip (cf. Defintion 4.1), for each image patch, we horizontally flipped it and add the flipped patch into the convolutional layer as a new filter. Thus, for an input CIFAR-10 image of size $3 2 \times 3 2$ , the dimension of the output feature is $8 1 9 2 \times 2 8 \times 2 8$ . To isolate the effect of randomness in the choices of the image patches, we fix the random seed to be 0 throughout the experiment. In this experiment, we set the value of the $c$ parameter in LAP to be $4 , 8 , 1 2 , \ldots , 2 0$ to avoid small and large values of $c$ . The results are reported in Tables 2 and 4. Similar to the experiments in Section 6.1, again we set the hyper-parameters by cross-validation, and the resulting accuracy is $8 8 . 9 1 \%$ . See Appendix $\mathrm { F }$ for the validation accuracy for different hyper-parameters.
+
+Table 1: Test accuracy of CNTK on CIFAR-10.   
+
+<table><tr><td>d C</td><td>5</td><td>8</td><td>11</td><td></td><td>14</td></tr><tr><td>0</td><td>66.55 (69.87)</td><td>66.27 (69.87)</td><td>65.85 (69.37)</td><td></td><td>65.47 (68.90)</td></tr><tr><td>4</td><td>77.06 (79.08)</td><td>77.14 (78.96)</td><td>77.06 (78.98)</td><td></td><td>76.52 (78.74)</td></tr><tr><td>8</td><td>79.24 (80.95)</td><td>79.25 (81.03)</td><td>78.98 (80.94)</td><td></td><td>78.65 (80.35)</td></tr><tr><td>12</td><td>80.11 (81.34)</td><td>79.79 (81.28)</td><td>79.29 (81.14)</td><td></td><td>79.13 (80.91)</td></tr><tr><td>16</td><td>79.80 (81.21)</td><td>79.71 (81.40)</td><td>79.74 (81.09)</td><td></td><td>79.42 (81.00)</td></tr><tr><td>20</td><td>79.24 (80.67)</td><td>79.27 (80.88)</td><td>79.30 (80.76)</td><td></td><td>78.92 (80.39)</td></tr><tr><td></td><td>78.07 (79.88)</td><td>78.16 (79.79)</td><td>78.14 (80.06)</td><td></td><td>77.87 (80.07)</td></tr><tr><td>28</td><td>76.91 (78.69)</td><td>77.33 (79.20)</td><td>77.65 (79.56)</td><td></td><td>77.65 (79.74)</td></tr><tr><td>32</td><td>76.79 (78.53)</td><td>77.39 (79.13)</td><td>77.63 (79.51)</td><td></td><td>77.63 (79.74)</td></tr></table>
+
+Table 2: Test accuracy of random patches layer $^ +$ CNTK on CIFAR-10.   
+
+<table><tr><td>d C</td><td colspan="2">5</td><td>8</td><td colspan="2">11</td><td colspan="2">14</td></tr><tr><td>4</td><td>84.63</td><td>(86.64) （</td><td>84.07 (86.23)</td><td></td><td>83.29 (85.53)</td><td></td><td>82.57 (84.81)</td></tr><tr><td>8</td><td>86.36(</td><td>(88.32)</td><td>85.80 (87.81)</td><td></td><td>85.01 (87.08)</td><td></td><td>84.57 (86.53)</td></tr><tr><td>12</td><td>86.74(</td><td>(88.35)</td><td>86.20 (87.90)</td><td></td><td>85.60 (87.36)</td><td></td><td>84.95 (86.99)</td></tr><tr><td>16</td><td></td><td>86.77 (88.36)</td><td>86.17 (87.85)</td><td></td><td>85.60 (87.44)</td><td></td><td>84.92 (86.98)</td></tr><tr><td>20</td><td>86.17(8</td><td>(87.77)</td><td>85.71 (87.50)</td><td></td><td>85.14 (87.07)</td><td></td><td>84.59 (86.84)</td></tr></table>
+
+From our experimental results, it is evident that combining CNTK or CNN-GP with additional feature extractor can significantly improve upon the performance of using solely CNTK or CNNGP, and that of using solely the feature extractor Coates et al. (2011). Previously, it has been reported in Recht et al. (2019) that using solely the feature extractor Coates et al. (2011) (together with appropriate pooling layer) can only achieve a test accuracy of $8 4 . 2 \%$ using 256, 000 image patches, or $8 3 . 3 \%$ using 32, 000 image patches. Even with the help of horizontal data augmentation, the feature extractor Coates et al. (2011) can only achieve a test accuracy of $8 5 . 6 \%$ using 256, $\ 0 0 0$ image patches, or $8 5 . 0 \%$ using 32, 000 image patches. Here we use significantly less image patches (only 2048) but achieve a much better performance, with the help of CNTK and CNN-GP. In particular, we achieve a performance of $8 8 . 9 1 \%$ on CIFAR-10, matching the performance of AlexNet on the same dataset. In the setting reported in Coates et al. (2011), increasing the number of sampled image patches will further improve the performance. Here we also conjecture that in our setting, further increasing the number of sampled image patches can improve the performance and get close to modern CNNs. However, due the limitation on computational resources, we leave exploring the effect of number of sampled image patches as a future research direction.
+
+# 6.3 EXPERIMENTS ON CNN WITH BOX FILTERING LAYER
+
+In Figure 1, we verify the effectiveness of BF on a 10-layer CNN (with Batch Normalization) on CIFAR-10. The setting of this experiment is reported in Appendix G. Our network structure has no pooling layer except for the BF layer before the last fully-connected layer. The fully-connected layer is fixed during the training. Our experiment illustrates that even with a fixed last FC layer, using
+
+Table 3: Test accuracy of CNN-GP on CIFAR-10.   
+
+<table><tr><td>d C</td><td>5</td><td>8</td><td>11</td><td></td><td>14</td></tr><tr><td>0</td><td>63.53 (67.90)</td><td>65.54 (69.43)</td><td>66.42 (70.30)</td><td></td><td>66.81 (70.48)</td></tr><tr><td>4</td><td>76.35 (78.79)</td><td>77.03 (79.30)</td><td>77.39 (79.52)</td><td></td><td>77.35 (79.65)</td></tr><tr><td>8</td><td>79.48 (81.32)</td><td>79.82 (81.49)</td><td>79.76 (81.71)</td><td></td><td>79.69 (81.53)</td></tr><tr><td>12</td><td>80.40 (82.13)</td><td>80.64 (82.09)</td><td>80.58 (82.06)</td><td></td><td>80.32 (81.95)</td></tr><tr><td>16</td><td>80.36 (81.73)</td><td>80.78 (82.20)</td><td>80.59 (82.06)</td><td></td><td>80.41 (81.83)</td></tr><tr><td>20</td><td>79.87 (81.50)</td><td>80.15( 5 (81.33)</td><td>79.87 (81.46)</td><td></td><td>79.98 (81.35)</td></tr><tr><td>24</td><td>78.60 (79.98)</td><td>78.91 (80.48)</td><td>79.22 (80.53)</td><td></td><td>78.94 (80.46)</td></tr><tr><td>28</td><td>77.18 (78.84)</td><td>78.03 (79.86)</td><td>78.45 (79.87)</td><td></td><td>78.48 (80.07)</td></tr><tr><td>32</td><td>77.00 (78.49)</td><td>77.85 (79.65)</td><td>78.49 (80.04)</td><td></td><td>78.45 (80.01)</td></tr></table>
+
+Table 4: Test accuracy of random patches layer $^ +$ CNN-GP on CIFAR-10.   
+
+<table><tr><td rowspan=1 colspan=1>dC</td><td rowspan=1 colspan=5>5                8                11              14</td></tr><tr><td rowspan=4 colspan=1>481216</td><td rowspan=1 colspan=1>85.49(87.32)</td><td rowspan=1 colspan=1>85.37(87.22)</td><td rowspan=1 colspan=1>85.16(87.11)</td><td rowspan=1 colspan=1>84.79(86.81)</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>87.07 (88.64)</td><td rowspan=1 colspan=1>86.82(88.68)</td><td rowspan=1 colspan=1>86.53(88.40)</td><td rowspan=2 colspan=2>86.39 (88.15)86.62 (88.29)</td></tr><tr><td rowspan=2 colspan=1>87.23(88.91)87.28 (88.90)</td><td rowspan=1 colspan=1>87.12(88.92)</td><td rowspan=1 colspan=1>86.87(88.66)</td></tr><tr><td rowspan=1 colspan=1>87.11(88.66)</td><td rowspan=1 colspan=1>86.92(88.61)</td><td rowspan=2 colspan=2>86.74 (88.24)86.26 (87.84)</td></tr><tr><td rowspan=1 colspan=1>20</td><td rowspan=1 colspan=1>86.81 (88.26)</td><td rowspan=1 colspan=1>86.77(88.24)</td><td rowspan=1 colspan=1>86.61(88.14)</td></tr></table>
+
+GAP could improve the performance of CNN. Our experiments also show that BF with appropriate choice of $c$ achieves better performance than GAP.
+
+![](images/320c235c4f1c90a590cb573560ef0fe6fc15d2b3b0c2ecdb55ed2f125f502004.jpg)  
+Figure 1: Test accuracy of 10-layer CNN with various values for the $c$ parameter in BF.
+
+# 7 CONCLUSION
+
+In this paper, inspired by the connection between full translation data augmentation and GAP, we derive a new operation, LAP, on CNTK and CNN-GP, which consistently improves the performance on image classification tasks. Combining CNN-GP with LAP and the pre-processing technique proposed by Coates et al. (2011), the resulting kernel achieves $89 \%$ accuracy on CIFAR-10, matching the performance of AlexNet and is the strongest classifier that is not a trained neural network.
+
+Here we list a few future research directions. Is it possible to develop analogs of CNTK or CNNGP incorporating modern techniques such as batch norm and residual layers, to further improve the performance? Moreover, it is an interesting direction to study other components in modern CNNs through the lens of CNTK and CNN-GP.
+
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+# A FORMAL DEFINITIONS OF CNN-GP AND CNTK
+
+We add some additional notations. Let $\pmb { I }$ be the identity matrix, and $[ n ] = \{ 1 , 2 , \dots , n \}$ . Let $e _ { i }$ be an indicator vector with $i$ -th entry being 1 and other entries being 0, and let 1 denote the all-one vector. We use $\odot$ to denote the pointwise product and $\otimes$ to denote the tensor product. We use $\mathrm { d i a g ( \cdot ) }$ to transform a vector to a diagonal matrix. We use $\sigma \left( \cdot \right)$ to denote the activation function, such as the rectified linear unit (ReLU) function: $\sigma \left( z \right) = \operatorname* { m a x } \{ z , 0 \}$ , and $\dot { \sigma } \left( \cdot \right)$ to denote the derivative of $\sigma \left( \cdot \right)$ . We set $c _ { \sigma } = 2$ . Denote by $\scriptstyle { \mathcal { N } } ( \mu , \Sigma )$ the Gaussian distribution with mean $\pmb { \mu }$ and covariance $\pmb { \Sigma }$ .
+
+Equation equation 2 shows patch $[ { \pmb w } * { \pmb x } ] _ { i j }$ depends on $\begin{array} { r } { [ { \pmb x } ] _ { i - \frac { q - 1 } { 2 } : i + \frac { q - 1 } { 2 } , j - \frac { q - 1 } { 2 } : j + \frac { q - 1 } { 2 } } } \end{array}$ . For $( i , j , i ^ { \prime } , j ^ { \prime } ) \in [ P ] \times [ Q ] \times [ P ] \times [ Q ]$ , define
+
+$$
+\begin{array} { r } { \mathfrak { I } _ { i j , i ^ { \prime } j ^ { \prime } } = \{ ( i + a , j + b , i ^ { \prime } + a ^ { \prime } , j ^ { \prime } + b ^ { \prime } ) \in [ P ] \times [ Q ] \times [ P ] \times [ Q ] | - ( q - 1 ) / 2 \le a , b , a ^ { \prime } , b ^ { \prime } \le ( q - 1 ) \} , } \end{array}
+$$
+
+Now we define the convolution operation. For a convolutional filter $\pmb { w } \in \mathbb { R } ^ { q \times q }$ and an image $\pmb { x } \in \mathbb { R } ^ { P \times Q }$ , the convolution operator is defined as
+
+$$
+[ { \pmb w } * { \pmb x } ] _ { i j } = \sum _ { a = - \frac { \eta - 1 } { 2 } } ^ { \frac { q - 1 } { 2 } } \sum _ { b = - \frac { \eta - 1 } { 2 } } ^ { \frac { q - 1 } { 2 } } [ { \pmb w } ] _ { a + \frac { q + 1 } { 2 } , b + \frac { q + 1 } { 2 } } [ { \pmb x } ] _ { a + i , b + j } \mathrm { ~ f o r ~ } i \in [ P ] , j \in [ Q ] .
+$$
+
+Now we formally define CNN.
+
+• Let $\pmb { x } ^ { ( 0 ) } = \pmb { x } \in \mathbb { R } ^ { P \times Q \times C ^ { ( 0 ) } }$ be the input image where $C ^ { ( 0 ) }$ is the initial number of channels. • For $h = 1 , \ldots , L , \beta = 1 , \ldots , C ^ { ( h ) }$ , the intermediate outputs are defined as
+
+$$
+\tilde { \mathbf { x } } _ { ( \beta ) } ^ { ( h ) } = \sum _ { \alpha = 1 } ^ { C ^ { ( h - 1 ) } } W _ { ( \alpha ) , ( \beta ) } ^ { ( h ) } * \mathbf { x } _ { ( \alpha ) } ^ { ( h - 1 ) } + \gamma \cdot b _ { ( \beta ) } , \quad \mathbf { x } _ { ( \beta ) } ^ { ( h ) } = \sqrt { \frac { c _ { \sigma } } { C ^ { ( h ) } \times q \times q } \sigma } \left( \tilde { \mathbf { x } } _ { ( \beta ) } ^ { ( h ) } \right)
+$$
+
+where each W (h) $W _ { ( \alpha ) , ( \beta ) } ^ { ( h ) } \in \mathbb { R } ^ { q \times q }$ is a filter with Gaussian initialization and $b _ { ( \beta ) }$ is a bias term with Gaussian initialization scaled by $\gamma$ .
+
+# CNN-GP and CNTK
+
+• For $\alpha = 1 , \dots , C ^ { ( 0 ) } , ( i , j , i ^ { \prime } , j ^ { \prime } ) \in [ P ] \times [ Q ] \times [ P ] \times [ Q ]$ , define
+
+$$
+\Big [ \Sigma ^ { ( 0 ) } ( { \pmb x } , { \pmb x } ^ { \prime } ) \Big ] _ { i j , i ^ { \prime } j ^ { \prime } } = \frac { 1 } { q ^ { 2 } } \sum _ { \alpha = 1 } ^ { C ^ { ( 0 ) } } \mathrm { t r } \left( \Big [ { \pmb K } _ { ( \alpha ) } ^ { ( 0 ) } ( { \pmb x } , { \pmb x } ^ { \prime } ) \Big ] _ { \mathscr D _ { i j , i ^ { \prime } j ^ { \prime } } } \right) + \beta ^ { 2 } .
+$$
+
+• For $h \in [ L ]$ , – For $( i , \dot { j } , i ^ { \prime } , j ^ { \prime } ) \in [ P ] \times [ Q ] \times [ P ] \times [ Q ]$ , define
+
+$$
+\begin{array} { r } { \pmb { \Lambda } _ { i j , i ^ { \prime } j ^ { \prime } } ^ { ( h ) } ( \pmb { x } , \pmb { x } ^ { \prime } ) = \left( \begin{array} { c c } { \left[ \pmb { \Sigma } ^ { ( h - 1 ) } ( \pmb { x } , \pmb { x } ) \right] _ { i j , i j } } & { \left[ \pmb { \Sigma } ^ { ( h - 1 ) } ( \pmb { x } , \pmb { x } ^ { \prime } ) \right] _ { i j , i ^ { \prime } j ^ { \prime } } } \\ { \left[ \pmb { \Sigma } ^ { ( h - 1 ) } \left( \pmb { x } ^ { \prime } , \pmb { x } \right) \right] _ { i ^ { \prime } j ^ { \prime } , i j } } & { \left[ \pmb { \Sigma } ^ { ( h - 1 ) } \left( \pmb { x } ^ { \prime } , \pmb { x } ^ { \prime } \right) \right] _ { i ^ { \prime } j ^ { \prime } , i ^ { \prime } j ^ { \prime } } } \end{array} \right) \in \mathbb { R } ^ { 2 \times 2 } . } \end{array}
+$$
+
+– Define ${ \pmb K } ^ { ( h ) } ( { \pmb x } , { \pmb x } ^ { \prime } ) , \dot { \pmb K } ^ { ( h ) } ( { \pmb x } , { \pmb x } ^ { \prime } ) \in \mathbb { R } ^ { P \times Q \times P \times Q }$ , for $( i , j , i ^ { \prime } , j ^ { \prime } ) \in [ P ] \times [ Q ] \times [ P ] \times [ Q ]$
+
+$$
+\begin{array} { r l } & { \left[ { K ^ { \left( h \right) } } ( { \pmb x } , { \pmb x } ^ { \prime } ) \right] _ { i j , i ^ { \prime } j ^ { \prime } } = c _ { \sigma } \cdot \underset { ( u , v ) \sim \mathcal { N } \left( \mathbf { 0 } , \Lambda _ { i j , i ^ { \prime } j ^ { \prime } } ^ { \left( h \right) } \left( \pmb x , { \pmb x } ^ { \prime } \right) \right) } { \mathbb { E } } \left[ \cdot \sigma \left( u \right) \sigma \left( v \right) \right] , } \\ & { \left[ \dot { \pmb K } ^ { ( h ) } ( { \pmb x } , { \pmb x } ^ { \prime } ) \right] _ { i j , i ^ { \prime } j ^ { \prime } } = c _ { \sigma } \cdot \underset { ( u , v ) \sim \mathcal { N } \left( \mathbf { 0 } , \Lambda _ { i j , i ^ { \prime } j ^ { \prime } } ^ { \left( h \right) } \left( \pmb x , { \pmb x } ^ { \prime } \right) \right) } { \mathbb { E } } \left[ \dot { \sigma } \left( u \right) \dot { \sigma } \left( v \right) \right] . } \end{array}
+$$
+
+– Define $\Sigma ^ { ( h ) } ( { \pmb x } , { \pmb x } ^ { \prime } ) \in \mathbb { R } ^ { P \times Q \times P \times Q }$ , for $( i , j , i ^ { \prime } , j ^ { \prime } ) \in [ P ] \times [ Q ] \times [ P ] \times [ Q ]$
+
+$$
+\left[ \Sigma ^ { ( h ) } ( \pmb { x } , \pmb { x } ^ { \prime } ) \right] _ { i j , i ^ { \prime } j ^ { \prime } } = \frac { 1 } { q ^ { 2 } } \mathrm { t r } \left( \left[ \pmb { K } ^ { ( h ) } ( \pmb { x } , \pmb { x } ^ { \prime } ) \right] _ { D _ { i j , i ^ { \prime } j ^ { \prime } } } \right) + \beta ^ { 2 } .
+$$
+
+Note that $\Sigma ( { \pmb x } , { \pmb x } ^ { \prime } )$ and $\dot { \Sigma } ( { \pmb x } , { \pmb x } ^ { \prime } )$ share similar structures as their NTK counterparts (Jacot et al., 2018). The only difference is that we have one more step, taking the trace over patches. This step represents the convolution operation in the corresponding CNN. Next, we can use a recursion to compute the final kernel value.
+
+1. First, we define ${ \Theta } ^ { ( 0 ) } ( x , x ^ { \prime } ) = \Sigma ^ { ( 0 ) } ( x , x ^ { \prime } )$ .   
+2. For $h = 1 , \ldots , L$ and $( i , j , i ^ { \prime } , j ^ { \prime } ) \in [ P ] \times [ Q ] \times [ P ] \times [ Q ]$ , we define
+
+$$
+\left[ \Theta ^ { ( h ) } ( \pmb { x } , \pmb { x } ^ { \prime } ) \right] _ { i j , i ^ { \prime } j ^ { \prime } } = \frac { 1 } { q ^ { 2 } } \mathrm { t r } \left( \left[ \dot { \pmb { K } } ^ { ( h ) } ( \pmb { x } , \pmb { x } ^ { \prime } ) \odot \Theta ^ { ( h - 1 ) } ( \pmb { x } , \pmb { x } ^ { \prime } ) + \pmb { K } ^ { ( h ) } ( \pmb { x } , \pmb { x } ^ { \prime } ) \right] _ { D _ { i j , i ^ { \prime } j ^ { \prime } } } \right) + \beta ^ { 2 } .
+$$
+
+# B ADDITIONAL DEFINITION AND PROOF FOR SECTION 4
+
+Definition B.1 (Group). $( { \mathcal { G } } , \circ )$ is $a$ group, if and only if
+
+1. each element $g \in { \mathcal { G } }$ is a operator: $\mathbb { R } ^ { P \times Q \times C }  \mathbb { R } ^ { P \times Q \times C }$ ;   
+2 $\ : \ : \forall g _ { 1 } , g _ { 2 } \in \mathcal { G } , g _ { 1 } \circ g _ { 2 } \in \mathcal { G } \ :$ , where $( g _ { 1 } \circ g _ { 2 } ) ( { \pmb x } )$ is defined as $g _ { 1 } ( g _ { 2 } ( { \pmb x } ) )$ .   
+3. $\forall g _ { 1 } , g _ { 2 } , g _ { 3 } \in \mathcal { G } , ( g _ { 1 } \circ g _ { 2 } ) \circ g _ { 3 } = g _ { 1 } \circ ( g _ { 2 } \circ g _ { 3 } ) .$ .   
+4. $\exists e \in { \mathcal { G } }$ , such that $\forall g \in { \mathcal { G } }$ , $e \circ g = g \circ e = g$ .   
+5. $\forall g _ { 1 } \in { \mathcal { G } }$ , $\exists g _ { 2 } \in { \mathcal { G } }$ , such that $g _ { 1 } \circ g _ { 2 } = g _ { 2 } \circ g _ { 1 } = e$ . We denote $g _ { 2 }$ as the inverse of $g _ { 1 }$ , namely,   
+$g _ { 1 } ^ { - 1 }$ .
+
+Proof of Theorem 4.1. Since we assume $\mathbf { K } _ { \mathbf { X } } ^ { \mathcal { G } }$ and $\mathbf { K } _ { \mathbf { X } _ { \mathcal { G } } }$ are invertible, both $_ { \pmb { \alpha } }$ and $\widetilde { \alpha }$ are uniquely defined. Now we claim $\widetilde { \pmb { \alpha } } _ { g } = \{ \widetilde { \alpha } _ { i , g } \} _ { i \in [ N ] } \in \mathbb { R } ^ { N }$ is equal to $\frac { \pmb { \alpha } } { | \mathscr { G } | }$ for all $g \in { \mathcal { G } }$ .
+
+By the invariance of $\mathbf { K }$ under $\mathcal { G }$ , for all $j \in [ N ]$ and $g ^ { \prime } \in \mathcal G$ ,
+
+$$
+\begin{array} { l } { { \displaystyle \sum _ { i \in [ N ] , g \in { \mathcal G } } \frac { \alpha _ { i } } { | { \mathcal G } | } \mathbf K ( g ^ { \prime } ( { \mathbf x } _ { j } ) , g ( { \mathbf x } _ { i } ) ) = \sum _ { i \in [ N ] , g \in { \mathcal G } } \frac { \alpha _ { i } } { | { \mathcal G } | } \mathbf K ( ( g ^ { - 1 } \circ g ^ { \prime } ) ( { \mathbf x } _ { j } ) , { \mathbf x } _ { i } ) } } \\ { ~ = \sum _ { i \in [ N ] } \alpha _ { i } \mathbb E _ { g \in { \mathcal G } } \mathbf K ( g ( { \mathbf x } _ { j } ) , { \mathbf x } _ { i } ) } \\ { ~ = \sum _ { i \in [ N ] } \alpha _ { i } \mathbf K ^ { { \mathcal G } } ( { \mathbf x } _ { j } , { \mathbf x } _ { i } ) } \\ { ~ = y _ { j } . } \end{array}
+$$
+
+Note that $\widetilde { \pmb { \alpha } }$ is defined as the unique solution of $\mathbf { K } _ { \mathbf { X } _ { \mathcal { G } } } \tilde { \pmb { \alpha } } = \pmb { y } _ { \mathcal { G } }$ , the claim has been verified.
+
+Similarly, we have
+
+$$
+\sum _ { i \in [ N ] , g \in \mathcal { G } } \frac { \alpha _ { i } } { | \mathcal { G } | } \mathbf { K } ( \pmb { x } ^ { \prime } , g ( \pmb { x } _ { i } ) ) = \sum _ { i \in [ N ] } \alpha _ { i } \mathbb { E } _ { g \in \mathcal { G } } \mathbf { K } ( g ^ { - 1 } ( \pmb { x } ^ { \prime } ) , \pmb { x } _ { i } ) = \sum _ { i \in [ N ] } \alpha _ { i } \mathbf { K } ^ { \mathcal { G } } ( \pmb { x } ^ { \prime } , \pmb { x } _ { i } ) .
+$$
+
+# C EQUIVALENCE BETWEEN LAP AND BOX FILTERING LAYER.
+
+For a CNN with a box filtering layer before the final fully-connected layer, the final output is defined $\begin{array} { r } { f ( \pmb { \theta } , \pmb { x } ) = \sum _ { \alpha = 1 } ^ { C ^ { ( L ) } } \Big \langle \pmb { W } _ { ( \alpha ) } ^ { ( L + 1 ) } , \mathsf { B F } \left( \pmb { W } _ { ( \alpha ) } ^ { ( L ) } \right) } \end{array}$ x(L)(α) E, where x(L)(α) $\pmb { x } _ { ( \alpha ) } ^ { ( L ) } \in \mathbb { R } ^ { P \times Q }$ , and ${ W _ { ( \alpha ) } ^ { ( L + 1 ) } } \in \mathbb { R } ^ { P \times Q }$ is the weight of the last fully-connected layer.
+
+Now we establish the equivalence between BF and LAP on CNTK. The equivalence on CNNGP can be derived similarly. Let $\Theta _ { \mathsf { B F } } \left( \pmb { x } , \pmb { x } ^ { \prime } \right) \ \in \ \mathbb { R } ^ { [ P ] \times [ Q ] \times [ P ] \times [ Q ] }$ be the CNTK kernel of BF $\left( \pmb { x } _ { ( \alpha ) } ^ { ( L ) } \right)$ . Since BF is just a linear operation, we have
+
+$$
+\left[ \Theta _ { \mathsf { B F } } \left( \pmb { x } , \pmb { x } ^ { \prime } \right) \right] _ { i , j , i ^ { \prime } , j ^ { \prime } } = \frac { 1 } { \left( 2 c + 1 \right) ^ { 4 } } \sum _ { \Delta _ { i } , \Delta _ { j } , \Delta _ { i ^ { \prime } } ^ { \prime } , \Delta _ { j ^ { \prime } } ^ { \prime } \in [ - c , c ] ^ { 4 } } \left[ \Theta ^ { \left( L \right) } \left( \pmb { x } , \pmb { x } ^ { \prime } \right) \right] _ { i + \Delta _ { i } , j + \Delta _ { j } , i ^ { \prime } + \Delta _ { i } ^ { \prime } , j ^ { \prime } + \Delta _ { j } ^ { \prime } } .
+$$
+
+By the formula of the output kernel value for CNTK without GAP, we obtain
+
+$$
+\mathrm { t r } \left( \Theta _ { \mathsf { B F } } \left( \pmb { x } , \pmb { x } ^ { \prime } \right) \right) = \frac { 1 } { \left( 2 c + 1 \right) ^ { 4 } } \sum _ { \Delta _ { i } , \Delta _ { i } ^ { \prime } , \Delta _ { j } , \Delta _ { j } ^ { \prime } \in [ - c , c ] ^ { 4 } } \sum _ { \substack { i , j \in [ P ] \times [ Q ] } } [ \Theta ( \pmb { x } , \pmb { x } ^ { \prime } ) ] _ { i + \Delta _ { i } , j + \Delta _ { j } , i + \Delta _ { i } ^ { \prime } , j + \Delta _ { j } ^ { \prime } } .
+$$
+
+![](images/1bbc9f757c12c5aed515820a88529545a710a5495ef9d4f4d35c646a539fd9a7.jpg)  
+Figure 2: Randomly sampled images with full translation data augmentation and local translation data augmentation from CIFAR-10. Full translation data augmentation can create unrealistic images that harm the performance whereas local translation data augmentation creates more realistic images.
+
+D EXPERIMENTAL SETUP IN SECTION 6
+
+For both CIFAR-10 and Fashion-MNIST we use the full training set and report the test accuracy on the full test set. Throughout this section we only consider $3 \times 3$ convolutional filters with stride 1 and no dilation. In the convolutional layers in CNTK and CNN-GP, we use zero padding with pad size 1 to ensure the input of each layer has the same size. We use zero padding for LAP throughout the experiment. We perform standard preprocessing (mean subtraction and standard deviation division) for all images.
+
+In all experiments, we perform kernel ridge regression to utilize the calculated kernel values4. We normalize the kernel matrices so that all diagonal entries are ones. Equivalently, we ensure all features have unit norm in RKHS. Since the resulting kernel matrices are usually ill-conditioned, we set the regularization term $\lambda = 5 \times 1 0 ^ { - 5 }$ , to make inverting kernel matrices numerically stable. We use one-hot encodings of the labels as regression targets. We use scipy.linalg.solve to solve the corresponding kernel ridge regression problem.
+
+The kernel value of CNTK and CNN-GP are calculated using the CuPy package. We write native CUDA codes to speed up the calculation of the kernel values. All experiments are performed on Amazon Web Services (AWS), using (possibly multiple) NVIDIA Tesla V100 GPUs. For efficiency considerations, all kernel values are computed with 32-bit precision.
+
+One unique advantage of the dynamic programming algorithm for calculating CNTK and CNNGP is that we do not need repeat experiments for, say, different values of $c$ in LAP and different depths. With our highly-optimized native CUDA codes, we spend roughly 1,000 GPU hours on calculating all kernel values for each dataset.
+
+# E TEST ACCURACY OF CNTK AND CNN-GP ON FASHION-MNIST
+
+Table 5: Test accuracy of CNTK on Fashion-MNIST.   
+
+<table><tr><td>d C</td><td colspan="3">5 8</td><td colspan="2">11</td></tr><tr><td>0</td><td>92.25 (92.56)</td><td>92.22</td><td>(92.51) 92.11</td><td>(92.29)</td><td>91.76 (92.17)</td></tr><tr><td>4</td><td>93.76 (94.07)</td><td>93.69 (93.86)</td><td>93.55</td><td>(93.74)</td><td>93.37 (93.58)</td></tr><tr><td>8</td><td>93.72 (93.96)</td><td>93.67 (93.78)</td><td>93.50</td><td>(93.58)</td><td>93.32 (93.51)</td></tr><tr><td>12</td><td>93.59 (93.80)</td><td>93.58 (93.70)</td><td>93.35</td><td>(93.44)</td><td>93.21 (93.40)</td></tr><tr><td>16</td><td>93.50 (93.62)</td><td>93.42 (93.63)</td><td>93.27</td><td>(93.40)</td><td>93.10 (93.25)</td></tr><tr><td>20</td><td>93.10 (93.34)</td><td>93.17 (93.49)</td><td></td><td>93.20 (93.34)</td><td>92.99 (93.18)</td></tr><tr><td>24</td><td>92.77 (93.04)</td><td>93.07 (93.44)</td><td>93.11 (</td><td>(93.31)</td><td>93.02 (93.21)</td></tr><tr><td>28</td><td>92.80 (92.98)</td><td>93.08 (93.42)</td><td>93.12(</td><td>(93.28)</td><td>92.97 (93.19)</td></tr></table>
+
+<table><tr><td>d C</td><td>5</td><td>8</td><td>11</td><td>14</td><td></td></tr><tr><td>0</td><td>91.47 (91.81)</td><td>91.96( (92.37)</td><td>92.09( (92.60)</td><td>92.22</td><td>(92.72)</td></tr><tr><td>4</td><td>93.44 (93.60)</td><td>93.59 (93.79)</td><td>93.63 (93.76)</td><td></td><td>93.59 (93.64)</td></tr><tr><td>8</td><td>93.26 (93.16)</td><td>93.41 (93.51)</td><td>93.31 (93.52)</td><td></td><td>93.39 (93.46)</td></tr><tr><td>12</td><td>92.83 3(92.94)</td><td>93.07 (93.20)</td><td>93.11 (93.15)</td><td></td><td>92.94 (93.09)</td></tr><tr><td>16</td><td>92.46 (92.51)</td><td>92.58 (92.83)</td><td>92.64( (92.92)</td><td></td><td>92.68 (93.07)</td></tr><tr><td>20</td><td>91.83 (91.72)</td><td>92.35 (92.42)</td><td>92.49 (92.79)</td><td></td><td>92.51 (92.69)</td></tr><tr><td>24</td><td>91.15 (91.40)</td><td>92.10 (92.18)</td><td>92.29 (92.60)</td><td></td><td>92.41 (92.77)</td></tr><tr><td>28</td><td>91.30 (91.37)</td><td>92.03 (92.27)</td><td>92.41( (92.79)</td><td></td><td>92.41 (92.74)</td></tr></table>
+
+Table 6: Test accuracy of CNN-GP on Fashion-MNIST.
+
+# F VALIDATION ACCURACY OF CNTK AND CNN-GP ON CIFAR-10 AND FASHION-MNIST
+
+<table><tr><td>d C</td><td colspan="2">5</td><td colspan="2">8</td><td colspan="2">11</td><td colspan="2">14</td></tr><tr><td>0</td><td>64.26</td><td>(68.42)</td><td>64.47 (68.23)</td><td></td><td>63.94 (67.80)</td><td></td><td></td><td>63.29 (67.00)</td></tr><tr><td>4</td><td></td><td>75.97 (78.87)</td><td>75.89 (78.99)</td><td></td><td>75.65 (78.56)</td><td></td><td></td><td>75.40 (78.19)</td></tr><tr><td>8</td><td></td><td>77.93 (80.65)</td><td>77.90 (80.69)</td><td></td><td>77.65 (80.41)</td><td></td><td></td><td>76.92 (79.94)</td></tr><tr><td>12</td><td></td><td>78.51 (80.73)</td><td></td><td>78.47 (80.85)</td><td>78.18 (80.57)</td><td></td><td>77.71 (80.19)</td><td></td></tr><tr><td>16</td><td></td><td>78.47 (80.39)</td><td></td><td>78.69 (80.56)</td><td>78.34 (80.17)</td><td></td><td></td><td>77.74 (79.97)</td></tr><tr><td>20</td><td></td><td>77.86 (79.69)</td><td>77.81 (79.81)</td><td></td><td>77.38 (79.55)</td><td></td><td></td><td>76.88 (79.46)</td></tr><tr><td>24</td><td></td><td>76.59 (78.12)</td><td>76.80 (78.63)</td><td></td><td>76.44 (78.79)</td><td></td><td></td><td>76.18 (78.73)</td></tr><tr><td>28</td><td></td><td>75.44 (77.08)</td><td>76.15 (78.20)</td><td></td><td></td><td>76.10 (78.30)</td><td></td><td>75.95 (78.37)</td></tr><tr><td>32</td><td></td><td>75.33 (76.99)</td><td></td><td>76.04 (78.09)</td><td></td><td>76.08 (78.27)</td><td></td><td>75.99 (78.32)</td></tr></table>
+
+Table 7: Validation accuracy of CNTK on CIFAR-10.
+
+<table><tr><td>d C</td><td>5</td><td>8</td><td>11</td><td></td><td>14</td></tr><tr><td>0</td><td>62.49 (66.63)</td><td>64.25 (68.20)</td><td>64.94 (69.01)</td><td></td><td>65.35 (69.29)</td></tr><tr><td>4</td><td>75.31 (78.59)</td><td>76.05 (79.20)</td><td>76.05 (79.17)</td><td></td><td>76.20 (79.09)</td></tr><tr><td>8</td><td>78.17 (81.02)</td><td>78.53 (81.29)</td><td>78.36 (81.20)</td><td></td><td>78.05 (80.98)</td></tr><tr><td>12</td><td>79.19 (81.38)</td><td>79.08 (81.66)</td><td>79.13 (81.52)</td><td></td><td>78.90 (81.10)</td></tr><tr><td>16</td><td>79.26 (81.18)</td><td>79.24 (81.37)</td><td>78.85 (81.33)</td><td></td><td>78.82 (80.84)</td></tr><tr><td>20</td><td>78.72 (80.61)</td><td>78.72 (80.85)</td><td>78.45 (80.60)</td><td></td><td>78.08 (80.22)</td></tr><tr><td></td><td>77.31 (79.01)</td><td>77.59 (79.49)</td><td>77.41 (79.56)</td><td></td><td>77.26 (79.38)</td></tr><tr><td>28</td><td>76.01 (77.60)</td><td>76.60 (78.32)</td><td>76.57 (78.76)</td><td></td><td>76.86 (79.01)</td></tr><tr><td>32</td><td>75.72 (77.54)</td><td>76.42 (78.47)</td><td>76.56 (78.94)</td><td></td><td>76.63 (78.87)</td></tr></table>
+
+Table 8: Validation accuracy of CNN-GP on CIFAR-10.
+
+<table><tr><td>d C</td><td colspan="2">5</td><td colspan="2">8</td><td colspan="2">14</td></tr><tr><td>4</td><td>83.89 (85.76)</td><td>83.13</td><td>(85.40)</td><td>82.62 (84.95)</td><td></td><td>82.02 (84.43)</td></tr><tr><td>8</td><td>85.52 (87.59)</td><td>84.88</td><td>(87.12)</td><td>84.30 (86.69)</td><td></td><td>83.84 (86.10)</td></tr><tr><td>12</td><td>85.71 (87.85)</td><td>85.32</td><td>(87.42)</td><td>84.81 (87.02)</td><td></td><td>84.26 (86.58)</td></tr><tr><td>16</td><td>85.68 (87.76)</td><td>85.19（</td><td>(87.30)</td><td>84.71 (86.83)</td><td></td><td>84.47 (86.40)</td></tr><tr><td>20</td><td>85.26 (87.11)</td><td>84.91</td><td>(86.67)</td><td>84.44 1(86.40)</td><td></td><td>84.09 (86.17)</td></tr></table>
+
+Table 9: Validation accuracy of additional feature extractor $^ +$ CNTK on CIFAR-10.
+
+<table><tr><td>d C</td><td>5</td><td>8</td><td>11</td><td>14</td></tr><tr><td>4</td><td>84.03 (86.16)</td><td>84.21 (86.38)</td><td>84.15 (86.33)</td><td>83.98 (86.04)</td></tr><tr><td>8</td><td>85.85 (87.94)</td><td>85.87 (88.03)</td><td>85.70 (87.87)</td><td>85.49 (87.62)</td></tr><tr><td>12</td><td>86.37 (88.33)</td><td>86.38 (88.25)</td><td>86.06 (88.12)</td><td>85.69 (87.82)</td></tr><tr><td>16</td><td>86.06 (88.27)</td><td>86.21 (88.05)</td><td>86.01 (87.87)</td><td>85.58 (87.74)</td></tr><tr><td>20</td><td>85.73 3(87.71)</td><td>85.79 (87.60)</td><td>85.73 (87.54)</td><td>85.27 (87.21)</td></tr></table>
+
+Table 10: Validation accuracy of additional feature extractor $^ +$ CNN-GP on CIFAR-10.
+
+<table><tr><td>d C</td><td>5</td><td>8</td><td>11</td><td></td><td>14</td></tr><tr><td>0</td><td>92.07 (92.30)</td><td>92.08 (92.21)</td><td>91.79 (91.99)</td><td></td><td>91.51 (91.72)</td></tr><tr><td>4</td><td>93.84 (93.96)</td><td>93.82 (93.83)</td><td>93.58 (93.74)</td><td></td><td>93.40 (93.57)</td></tr><tr><td>8</td><td>93.82 (93.96)</td><td>93.80 (93.77)</td><td>93.56 (93.71)</td><td></td><td>93.37 (93.57)</td></tr><tr><td>12</td><td>93.71 (93.83)</td><td>93.60 (93.72)</td><td>93.45</td><td>(93.58)</td><td>93.41 (93.45)</td></tr><tr><td>16</td><td>93.59 (93.73)</td><td>93.39 (93.63)</td><td>93.34 (93.53)</td><td></td><td>93.21 (93.45)</td></tr><tr><td>20</td><td>93.24 (93.44)</td><td>93.29 (93.42)</td><td>93.26 (93.30)</td><td></td><td>93.19 (93.31)</td></tr><tr><td>24</td><td>93.16 (93.28)</td><td>93.21 (93.39)</td><td>93.30 (93.32)</td><td></td><td>93.22 (93.32)</td></tr><tr><td>28</td><td>93.11 (93.23)</td><td>93.21 (93.33)</td><td>93.29 ( (93.29)</td><td></td><td>93.28 (93.31)</td></tr></table>
+
+Table 11: Validation accuracy of CNTK on Fashion-MNIST.
+
+<table><tr><td>d C</td><td colspan="2">5</td><td colspan="2">8</td><td colspan="2">14</td></tr><tr><td>0</td><td>91.13 (91.43)</td><td>91.57</td><td>(91.77)</td><td>91.85 (91.92)</td><td></td><td>91.94 (92.08)</td></tr><tr><td>4</td><td>93.44 (93.55)</td><td></td><td>93.57 (93.54)</td><td>93.69 (93.68)</td><td></td><td>93.58 (93.64)</td></tr><tr><td>8</td><td>93.57 (93.67)</td><td></td><td>93.51 (93.68)</td><td>93.52 (93.72)</td><td></td><td>93.44 (93.58)</td></tr><tr><td>12</td><td>93.15( (93.36)</td><td></td><td>93.49 (93.59)</td><td>93.25 (93.52)</td><td></td><td>93.23 (93.44)</td></tr><tr><td>16</td><td>92.83 (92.84)</td><td></td><td>93.01 (93.19)</td><td>93.01 (93.27)</td><td></td><td>92.95 (93.18)</td></tr><tr><td>20</td><td>92.29 (92.45)</td><td></td><td>92.60 (92.82)</td><td>92.60 (92.93)</td><td></td><td>92.78 (93.10)</td></tr><tr><td>24</td><td>91.76 (92.04)</td><td></td><td>92.28 (92.63)</td><td>92.57 (92.86)</td><td></td><td>92.58 (92.78)</td></tr><tr><td>28</td><td>91.79 (92.00)</td><td></td><td>92.32 (92.56)</td><td>92.56 (92.77)</td><td></td><td>92.70 (93.00)</td></tr></table>
+
+Table 12: Validation accuracy of CNN-GP on Fashion-MNIST.
+
+# G SETTING OF THE EXPERIMENT IN SECTION 6.3
+
+The total number of training epochs is 80, and the learning rate is 0.1 initially, decayed by 10 at epoch 40 and 60 respectively. The momentum is 0.9 and the weight decay factor is 0.0005. In Figure 1, the blue line reports the average test accuracy of the last 10 epochs, while the red line reports the best test accuracy of the total 80 epochs. Each experiment is repeated for 3 times. We use circular padding for both convolutional layers and the BF layer. The last data point with largest $x$ -coordinate reported in Figure 1 corresponds to GAP.

parse/train/BylIciRcYQ/BylIciRcYQ.md

@@ -0,0 +1,362 @@
+# SGD CONVERGES TO GLOBAL MINIMUM IN DEEP LEARNING VIA STAR-CONVEX PATH
+
+Yi Zhou∗, Junjie Yang†, Huishuai Zhang‡, Yingbin Liang§, Vahid Tarokh∗ ∗Duke University, †University of Science and Technology of China ‡Microsoft Research, Asia, §The Ohio State University
+
+# ABSTRACT
+
+Stochastic gradient descent (SGD) has been found to be surprisingly effective in training a variety of deep neural networks. However, there is still a lack of understanding on how and why SGD can train these complex networks towards a global minimum. In this study, we establish the convergence of SGD to a global minimum for nonconvex optimization problems that are commonly encountered in neural network training. Our argument exploits the following two important properties: 1) the training loss can achieve zero value (approximately), which has been widely observed in deep learning; 2) SGD follows a star-convex path, which is verified by various experiments in this paper. In such a context, our analysis shows that SGD, although has long been considered as a randomized algorithm, converges in an intrinsically deterministic manner to a global minimum.
+
+# 1 INTRODUCTION
+
+Training neural networks has been proven to be NP-hard decades ago Blum & Rivest (1988). At that time, the limited computation power makes neural network training a “mission impossible”. However, as the development of computing device achieves several revolutionary milestones (e.g. GPUs), deep neural networks are found to be practically trainable and can generalize well on real datasets Krizhevsky et al. (2017). At the same time, deep learning technique starts to beat the performance of other conventional approaches in a variety of challenging tasks, e.g., computer vision, classification, natural language processing, etc.
+
+Modern neural network training is typically performed by applying first-order algorithms such as stochastic gradient descent (SGD) (a.k.a backpropagation) (Linnainmaa, 1976) or its variants, e.g., Adam (Kingma & Ba, 2014), Adagrad (Duchi et al., 2011), etc. Traditional analysis of SGD in nonconvex optimization guarantees the convergence to a stationary point Bottou et al. (2016); Ghadimi et al. (2016). Recently, it has been shown that SGD has the capability to escape strict saddle points Ge et al. (2015); Jin et al. (2017); Reddi et al. (2018a); Daneshmand et al. (2018), and can escape even sharp local minima Kleinberg et al. (2018). While these works provide different insights towards understanding the performance of SGD, they cannot explain the success of SGD that has been widely observed in deep learning applications. Specifically, it is known that SGD is able to train a variety of deep neural networks to achieve zero training loss (either exactly or approximately) for non-negative loss functions. This implies that SGD can find a global minimum of deep neural networks at ease. The major challenges towards understanding this phenomenon are in two-fold: 1) deep neural networks have complex landscape that cannot be fully understood analytically (Zhou & Liang, 2018); 2) the randomness nature of SGD makes it hard to characterize its convergence on such a complex landscape. These factors prohibit a good understanding of the practical success of SGD in deep learning from a traditional optimization aspect.
+
+In this study, we analyze the convergence of SGD in the above phenomenon by exploiting the following two critical properties. First, the fact that SGD can train neural networks to zero loss value implies that the non-negative loss functions on all data samples share a common global minimum. Second, our experiments establish strong empirical evidences that SGD (when training the loss to zero value) follows a star-convex path. Based on these properties, we formally establish the convergence of SGD to a global minimizer. Our work conveys a useful insight that although the landscape of neural networks can be complicated, the actual optimization path that SGD takes turns out to be remarkably simple and sufficient to guarantee the converge to a global minimum.
+
+# 1.1 OUR CONTRIBUTIONS
+
+We focus on the empirical observation that SGD can train various neural networks to achieve zero loss, and validate that SGD follows an epochwise star-convex path in empirical optimization processes. Based on such a property, we prove that the Euclidean distance between the variable sequence generated by SGD and a global minimizer decreases at an epoch level. We also show that the subsequences of iterations that correspond to the same data sample is a minimizing sequence of the loss corresponding to that sample.
+
+By further empirical exploration, we validate that SGD follows an iterationwise star-convex path during the major part of the training process. Based on such a property, we prove that the entire variable sequence generated by SGD converges to a global minimizer of the objective function. Then, we show that the convergence of SGD induces a self-regularization on its variance, i.e., the variance of stochastic gradients vanishes as SGD converges in such a context.
+
+From a technical perspective, we characterize the intrinsic deterministic convergence property of SGD when the optimization path is well regularized, rather than the performance on average or in probability established in the existing studies. Our results provide a novel and promising aspect to understand SGD-based optimization in deep learning. Furthermore, our analysis of SGD explores the limiting convergence property of the subsequences that correspond to individual data samples, which is in sharp contrast to the traditional treatment of SGD that depends on its random nature and bounds on variance. Hence, our proof technique can be of independent interest to the community.
+
+# 1.2 RELATED WORK
+
+As there are extensive literature on SGD, we only mention the highly relevant studies here. Theoretical foundations of SGD have been developed in the optimization community (Robbins & Monro, 1951; Nemirovski et al., 2009; Lan, 2012; Ghadimi et al., 2016; Ghadimi & Lan, 2016), and have attracted much attention from the machine learning community in the past decade (Schmidt et al., 2017; Defazio et al., 2014; Johnson & Zhang, 2013; Li et al., 2017; Wang et al., 2018a). In general nonconvex optimization, it is known that SGD converges to a stationary point under a bounded variance assumption and a diminishing learning rate Bottou et al. (2016); Ghadimi et al. (2016). Other variants of SGD that are designed for deep learning have been proposed, e.g., Adam (Kingma & Ba, 2014), AMSgrad Reddi et al. (2018b), Adagrad (Duchi et al., 2011), etc, and their convergence properties have been studied in the context of online convex regret minimization.
+
+Needell et al. (2014); Moulines & Bach (2011) show that SGD converges at a linear rate when the objective function is strongly convex and has a unique common global minimum. Recently, several studies show that SGD has the capability to escape strict saddle points Ge et al. (2015); Jin et al. (2017); Reddi et al. (2018a); Daneshmand et al. (2018). Other cubic-regularization-based methods have also been shown to be able to escape strict saddle points Nesterov & Polyak (2006); Zhou et al. (2018); Wang et al. (2018b). Kleinberg et al. (2018) considers functions with one-point strong convexity, and shows that the randomness of SGD has an intrinsic smoothing effect that can avoid convergence to sharp minimum. Our paper exploits a very different notion of star-convexity path of SGD, which is a much weaker condition than those in the previous studies.
+
+# 2 PROBLEM SETUP AND PRELIMINARIES
+
+Neural network training can be formulated as the following finite-sum optimization problem.
+
+$$
+\operatorname* { m i n } _ { x \in \mathbb { R } ^ { d } } f ( x ) : = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell _ { i } ( x ) ,
+$$
+
+where there are in total $n$ training data samples. The loss function that corresponds to the $i$ -th data sample is denoted by $\ell _ { i } : \mathbb { R } ^ { \breve { d } } \to \mathbb { R }$ for $i = 1 , \ldots , n$ , and the vector to be minimized in the problem (e.g., network weights in deep learning) are denoted by $x \in \mathbb { R } ^ { d }$ . In general, problem $( \mathrm { P } )$ is a nonconvex optimization problem, and we make the following standard assumptions regarding the objective function.
+
+Assumption 1. The loss functions $\ell _ { i } , i = 1 , \ldots , n$ in problem $( P )$ satisfy:
+
+1. They are continuously differentiable, and their gradients are $L$ -Lipschitz continuous;
+
+2. For every $i = 1 , \dots , n , \operatorname* { i n f } _ { x \in \mathbb { R } ^ { d } } \ell _ { i } ( x ) > 0 .$
+
+The conditions imposed by Assumption 1 are standard in analysis of nonconvex optimization. In specific, item 1 is a standard smoothness assumption on nonconvex loss functions. Item 2 assumes that the loss functions are bounded below, which is satisfied by many loss functions in deep learning, e.g., MSE loss, crossentropy loss, NLL loss, etc, which are all non-negative.
+
+Next, we introduce the following fact that is widely observed in training overparameterized deep neural networks, which we assume to hold throughout the paper.
+
+Observation 1 (Global minimum in deep learning). The objective function $f$ in problem $( P )$ with non-negative loss can achieve zero value at certain $x ^ { * }$ . Thus, $x ^ { * }$ is also a common global minimizer for all individual loss $\{ \ell _ { i } \} _ { i = 1 } ^ { n }$ . More formally, denote $\mathcal { X } _ { i } ^ { \ast }$ as the set of global minimizers of $\ell _ { i }$ for $i = 1 , \ldots , n$ . Then, the set of common global minimizers, i.e., $\mathcal { X } ^ { * } : = \cap _ { i = 1 } ^ { n } \mathcal { X } _ { i } ^ { * }$ , is non-empty and bounded.
+
+Observation 1 is a common observation in deep learning applications, because deep neural networks (especially in the overparameterized regime) typically have enough capacity to fit all training data samples, and therefore the model has a global minimum shared by the loss functions on all data samples. To elaborate more, if the total loss $f$ attains zero value at $x ^ { * }$ , then $\ell _ { i }$ for all $i$ must achieve zero value at $x ^ { * }$ as they are non-negative. Thus, $x ^ { * }$ is a common global minimum of all individual loss $\ell _ { i } ( x )$ for all $i$ . Such a fact plays a critical role in understanding the convergence of SGD in training neural networks.
+
+Next, we introduce our algorithm of interest – stochastic gradient descent (SGD). Specifically, to solve problem (P), SGD starts at an initial vector $x _ { 0 } \in \mathbb { R } ^ { d }$ and generates a variable sequence $\{ x _ { k } \} _ { k }$ according to the following update rule.
+
+$$
+\begin{array} { r l } { \mathrm { ( S G D ) } } & { { } x _ { k + 1 } = x _ { k } - \eta \nabla \ell _ { \xi _ { k } } ( x _ { k } ) , \quad k = 0 , 1 , . . . , } \end{array}
+$$
+
+where $\eta > 0$ denotes the learning rate, and $\xi _ { k } \in \{ 1 , \ldots , n \}$ corresponds to the sampled data index at iteration $k$ . In this study, we consider the practical cyclic sampling scheme with reshuffle (also referred to as the random sampling without replacement) to generate the random variable $\xi _ { k }$ . To elaborate the notation, we rewrite every iteration number $k$ as $n B + t$ , where $B = 0 , 1 , 2 , \ldots$ denotes the index of epoch that iteration $k$ belongs to and $t \in \{ 0 , 1 , . . . , n - 1 \}$ denotes the corresponding iteration index in that epoch. We further denote $\pi _ { B }$ as the random permutation of $1 , . . . , n$ in the $B$ -th epoch, and denote $\pi _ { B } ( j )$ as its $j$ -th element. Then, the sampled data index at iteration $k$ can be expressed as
+
+$$
+\xi _ { k } = \pi _ { B } ( t + 1 ) , \quad t = 0 , . . . , n - 1 .
+$$
+
+# 3 APPROACHING GLOBAL MINIMUM EPOCHWISELY
+
+# 3.1 AN INTERESTING EMPIRICAL OBSERVATION OF SGD PATH
+
+In this subsection, we provide empirical observations on the algorithm path of SGD in training neural networks. To be specific, we train a standard multi-layer perceptron (MLP) network Krizhevsky (2009), a variant of Alexnet and a variant of Inception network Zhang et al. (2017a) on the CIFAR10 dataset Krizhevsky (2009) using SGD under crossentropy loss. In all experiments, we adopt a constant learning rate (0.01 for MLP and Alexnet, 0.1 for Inception) and a constant mini-batch size 128. We discard all other optimization features such as momentum, weight decay, dropout and batch normalization, etc, in order to observe the essential property of SGD.
+
+In each experiment, we train the network for a sufficient number of epochs to achieve near-zero training loss (i.e., almost global minimum), and record the weight parameters along the iteration path of SGD. We denote the weight parameters produced by SGD in the last iteration as $x ^ { * }$ , which has a near zero loss, and evaluate the Euclidean distance between the weight parameters produced by SGD and $x ^ { * }$ along the iteration path. We plot the results in Figure 1. It can be seen that the training losses for all three networks fluctuate along the iteration path, implying that the algorithm passes through complex landscapes. However, the Euclidean distance between the weight parameters and the final output $x ^ { * }$ is monotonically decreasing epochwise along the SGD path for all three networks. This shows that the variable sequence generated by SGD approaches the global minimum $x ^ { * }$ in a remarkably stable way. This motivates us to explore the underlying mechanism that yields such interesting observations.
+
+In the next two subsections, we first propose a property that the algorithm path of SGD satisfies, based on which we formally prove that the variable sequence generated by SGD admits the behavior observed in Figure 1. Then, we provide empirical evidences to validate such a property of SGD path in practical SGD training.
+
+![](images/c122fe5d26fdcae1410b98a7230ad63527b055c7fd9264d47dc9bc4cb09ff6a1.jpg)  
+Figure 1: Distance to output of SGD in training neural networks.
+
+# 3.2 EPOCHWISE STAR-CONVEX PATH
+
+In this subsection, we introduce the notion of the epochwise star-convex path for SGD and establish its theoretical implications on the convergence of SGD. We validate that SGD satisfies such a property in practical neural network training in Section 3.3.
+
+Recall the conventional definition of star-convexity. Let $x ^ { * }$ be a global minimizer of a smooth function $h$ . Then, $h$ is said to be star-convex at a point $x$ provided that
+
+$$
+( { \mathrm { S t a r - c o n v e x i t y } } ) : \quad h ( x ) - h ( x ^ { * } ) + \langle x ^ { * } - x , \nabla h ( x ) \rangle \leq 0 .
+$$
+
+Star-convexity can be intuitively understood as convexity between a reference point $x$ and a global minimizer $x ^ { * }$ . Such a property ensures that the negative gradient $- \nabla h ( x )$ points to the desired direction $x ^ { * } - x$ for minimization.
+
+Next, we define the notion of epochwise star-convex path, which requires the star-convexity to be held cumulatively over each epoch.
+
+Definition 1 (Epochwise star-convex path). We call a path generated by SGD epochwise star-convex if it satisfies: For all epochs $B = 0 , 1 , \ldots$ and for a fixed $x ^ { \ast } \in \mathcal { X } ^ { \ast }$ (see Observation 1 for definition),
+
+$$
+\sum _ { k = n B } ^ { n ( B + 1 ) - 1 } \left[ \ell _ { \xi _ { k } } ( x _ { k } ) - \ell _ { \xi _ { k } } ( x ^ { * } ) + \langle x ^ { * } - x _ { k } , \nabla \ell _ { \xi _ { k } } ( x _ { k } ) \rangle \right] \leq 0 .
+$$
+
+We note that the property introduced by Definition 1 is not about the landscape geometry of a loss function, which can be complex as observed in the training loss curves shown in Figure 1. Rather, it characterizes the interaction between the algorithm and the loss function along the optimization path. Such a property is generally weaker than the global star-convexity, and is observed to be held in practical neural network training (see Section 3.3).
+
+Based on Definition 1, we obtain the following property of SGD.
+
+Theorem 1 (Epochwise diminishing distance). Let Assumption 1 hold and apply SGD with learning rate $\begin{array} { r } { \eta < { \frac { 1 } { L } } } \end{array}$ to solve problem $( P )$ . Assume SGD follows an epochwise star-convex path for a certain $x ^ { * } \in \mathcal { X } ^ { * }$ . Then, the variable sequence $\{ x _ { k } \} _ { k }$ generated by SGD satisfies, for all epochs $B = 0 , 1 , . . . ,$
+
+$$
+\| x _ { n ( B + 1 ) } - x ^ { * } \| \leq \| x _ { n B } - x ^ { * } \| .
+$$
+
+Theorem 1 proves that the variable sequence generated by SGD approaches a global minimizer at an epoch level, which is consistent with the empirical observations made in Figure 1. Therefore, the property of epochwise star-convex path of SGD is sufficient to explain such desirable empirical observations, although the loss function can be highly nonconvex and has complex landscape.
+
+Under the cyclic sampling scheme with reshuffle, SGD samples every data sample once per epoch. Consider the loss $\ell _ { v }$ on the $v$ -th data sample for a fixed $v \in \{ 1 , 2 , . . . , n \}$ . One can check that the iterations in which the loss $\ell _ { v }$ is sampled form a subsequence $\{ x _ { n B + \pi _ { B } ^ { - 1 } ( v ) } \} _ { B }$ , where $\pi _ { B } ^ { - 1 }$ is the inverse permutation mapping of $\pi _ { B }$ , i.e., $\pi _ { B } ^ { - 1 } ( u ) = v$ if and only if $\pi _ { B } ( v ) = u$ . Next, we characterize the convergence properties of these subsequences corresponding to the loss functions on individual data samples.
+
+Theorem 2 (Minimizing subsequences). Under the same settings as those of Theorem 1, the subsequences $\{ x _ { n B + \pi _ { B } ^ { - 1 } ( v ) } \} _ { B }$ for $v = 1 , . . . , n$ satisfy
+
+1. They are minimizing sequences for the corresponding loss functions, i.e.,
+
+$$
+\operatorname* { l i m } _ { B \to \infty } \ell _ { v } ( x _ { n B + \pi _ { B } ^ { - 1 } ( v ) } ) = \operatorname* { i n f } _ { x \in \mathbb { R } ^ { d } } \ell _ { v } ( x ) , \quad \forall v \in \{ 1 , . . . , n \} .
+$$
+
+2. Every limit point of $\{ x _ { n B + \pi _ { B } ^ { - 1 } ( v ) } \} _ { B }$ is in $\mathcal { X } _ { v } ^ { \ast }$
+
+Theorem 2 characterizes the limiting behavior of the subsequences that correspond to the loss functions on individual data samples. Essentially, the results in items 1 and 2 show that each subsequence $\{ x _ { n B + \pi _ { B } ^ { - 1 } ( v ) } \} _ { B }$ is a minimizing sequence for the corresponding loss $\ell _ { v }$ .
+
+# DISCUSSION
+
+We note that Theorems 1 and 2 characterize the epochwise convergence property of SGD in a deterministic way. The underlying technical reason is that the common global minimizer structure in Observation 1 suppresses the randomness induced by sampling and reshuffling of SGD, and ensures a common direction along which SGD can approach the global minimum on all individual data samples. Such a result is very different from traditional understanding of SGD where randomness and variance play a central role Nemirovski et al. (2009); Ghadimi et al. (2016).
+
+# 3.3 VERIFYING EPOCHWISE STAR-CONVEX PATH OF SGD
+
+In this subsection, we conduct experiments to validate the epochwise star-convex path of SGD introduced in Definition 1.
+
+We train the aforementioned three types of neural networks, i.e., MLP, Alexnet and Inception, on CIFAR10 Krizhevsky (2009) and MNIST Lecun et al. (1998) dataset using SGD. The hyperparameter settings are the same as those mentioned in Section 3.1. We train these networks for a sufficient number of epochs to achieve a near-zero training loss (i.e., near-global minimum). We record the variable sequence generated by SGD along the entire algorithm path, and set $x ^ { * }$ to be the final output of SGD. Then, we evaluate the value of the summation term in Definition 1 for each epoch. The value of this summation term for each epoch $B$ is denoted as residual $e _ { B }$ . By Definition 1, SGD path in the $B$ -th epoch is epochwise star-convex provided that $e _ { B } < 0$ .
+
+Figure 2 shows the results of our experiments. In all subfigures, the red horizontal curve denotes the zero value baseline, and the other curve denotes the residual $e _ { B }$ . It can be seen from Figure 2 that, on the MNIST dataset (second row), the entire path of SGD satisfies epochwise star-convexity for all three networks. On the CIFAR10 dataset (first row), we observe an epochwise star-convex path of SGD after several epochs of the initial phase of training. This can be due to the more complex landscape of the loss function on the CIFAR10 dataset, so that it takes SGD several epochs to enter a basin of attraction of the global minimum.
+
+Our empirical findings strongly support the validity of the epochwise star-convex path of SGD in Definition 1. Therefore, Theorem 1 establishes an empirically-verified theory for characterizing the convergence property of SGD in training neural networks at an epoch level. In particular, it is well justified to successfully explain the stable epochwise convergence behavior observed in Figure 1.
+
+![](images/504d72f48c6e375be614122b28d35ba6f006dbaf2772a5ec1de37cc2abbedbc6.jpg)  
+Figure 2: Verification of epochwise star-convex path.
+
+# 4 CONVERGENCE TO A GLOBAL MINIMIZER
+
+The result developed in Theorem 1 shows that the variable sequence generated by SGD monotonically approaches a global minimizer at an epoch level. However, it does not guarantee the convergence of the variable sequence to a global minimizer (which requires the distance between SGD iterates and the global minimizer reduces to zero). We further explore such a convergence issue in the following two subsections. We first define a notion of an iterationwise star-convex path for SGD, based on which we formally establish the convergence of SGD to a global minimizer. Then, we provide empirical evidences to support the satisfaction of the iterationwise star-convex path by SGD.
+
+# 4.1 ITERATIONWISE STAR-CONVEX PATH
+
+We introduce the following definition of an iterationwise star-convex path for SGD.
+
+Definition 2 (Iterationwise star-convex path). We call a path generated by SGD iterationwise starconvex if it satisfies: For all $k = 0 , 1 , \ldots$ and for every ${ \boldsymbol { x } } ^ { * } \in \mathcal { X } _ { \xi _ { k } } ^ { * }$ ,
+
+$$
+\ell _ { \xi _ { k } } ( x _ { k } ) - \ell _ { \xi _ { k } } ( x ^ { * } ) + \langle x ^ { * } - x _ { k } , \nabla \ell _ { \xi _ { k } } ( x _ { k } ) \rangle \leq 0 .
+$$
+
+Compared to Definition 1 which defines the star-convex path of SGD at an epoch level, Definition 2 characterizes the star-convexity of SGD along the optimization path at a more refined iteration level. As we show in the result below, such a stronger property helps to regularize the convergence property of SGD at an iteration level, and is sufficient to guarantee convergence.
+
+Theorem 3 (Convergence to global minimizer). Let Assumption 1 hold and apply SGD with learning rate $\begin{array} { r } { \eta < { \frac { 1 } { L } } } \end{array}$ to solve problem $( P )$ . Assume SGD follows an iterationwise star-convex path. Then, the sequence $\{ x _ { k } \} _ { k }$ generated by SGD converges to a global minimizer.
+
+Theorem 3 formally establishes the convergence of SGD to a global minimizer along an iterationwise star-convex path. The main idea of the proof is to establish a consensus of the minimizing subsequences that are studied in Theorem 2, i.e., all these subsequences converge to the same limit – a common global minimizer of the loss functions over all the data samples. More specifically, our proof strategy consists of three steps: 1) show that every limit point of each subsequence is a common global minimizer; 2) prove that each subsequence has a unique limit point; 3) show that all these subsequences share the same unique limit point, which is a common global minimizer. We believe that the proof technique here can be of independent interest to the community.
+
+Our analysis in Theorem 3 characterizes the intrinsic deterministic convergence property of SGD, which is an alternative view of the SGD path: It performs gradient descent on an individual loss component at each iteration. The star-convexity along the iteration path pushes the algorithm towards the common global minimizer. Such progress is shared across all data samples in every iteration and eventually leads to the convergence of SGD.
+
+We also note that the convergence result in Theorem 3 is based on a constant learning rate, which is typically used in practical training. This is very different from and much more desirable than the diminishing learning rate adopted in traditional analysis of SGD Nemirovski et al. (2009), which is a necessity to mitigate the negative effects caused by the variance of SGD. Furthermore, Theorem 3 shows that SGD converges to a common global minimizer where the gradient of loss function on all data samples vanish, and we therefore obtain the following interesting corollary.
+
+Corollary 1 (Vanishing variance). Under the same settings of those of Theorem 3, the variance of stochastic gradients sampled by SGD converges to zero as iteration k goes to infinity.
+
+Thus, upon convergence, the common global minimizer structure in deep learning leads to a selfvariance-reducing effect on SGD. Such a desirable effect is the core property of stochastic variancereduced algorithms that reduces sample complexity Johnson & Zhang (2013). Hence, this justifies in part that SGD is a sample-efficient algorithm in learning deep models.
+
+# DISCUSSION
+
+We want to mention that many nonconvex sensing models have an underlying true signal and hence naturally have common global minimizers, e.g., phase retrieval Zhang et al. (2017b), low-rank matrix recovery Tu et al. (2016), blind deconvolution Li et al. (2018), etc. This is also the case for some neural network sensing problems Zhong et al. (2017). Also, these problems have been shown to satisfy the so-called gradient dominance condition and the regularity condition locally around the global minimizers Zhou et al. (2016); Tu et al. (2016); Li et al. (2018); Zhong et al. (2017); Zhou & Liang (2017). These two geometric properties imply the star-convexity of the objective function, which necessarily imply the epochwise and iterationwise star-convex path of SGD. Therefore, our results also have implications on the convergence guarantee of SGD for solving these problems as well.
+
+![](images/a9aa078b572784a7e6a40bb7aee0272a907ca6a07fd29593831b62eadcfa463b.jpg)  
+Figure 3: Verification of iterationwise star-convex path under crossentropy loss.
+
+# 4.2 VERIFYING ITERATIONWISE STAR-CONVEX PATH OF SGD
+
+In this subsection, we conduct experiments to validate the iterationwise star-convex path of SGD introduced in Definition 2.
+
+We train the aforementioned three types of neural networks, i.e., MLP, Alexnet and Inception, on CIFAR10 and MNIST datasets using SGD. The hyperparameter settings are the same as those mentioned in Section 3.1. We train these networks for a sufficient number of epochs to achieve a near-zero training loss. Due to the demanding requirement for storage, we record the variable sequence generated by SGD for all iterations in every tenth epoch, and set $x ^ { * }$ to be the final output of SGD. Then, for all the iterations in every tenth epoch, we evaluate the corresponding values of the terms on the left hand side in eq. (2) (denoted as $e _ { k }$ ). Then, we report the fraction of number of iterations that satisfy the iterationwise star-convexity (i.e., $e _ { k } < 0$ ) within such an epoch.
+
+In all subfigures of Figure 3, the red curves denote the training loss and the blue bars denote the fraction of iterations that satisfy the iterationwise star-convexity within such an epoch. It can be seen from Figure 3 that, for all three networks on the MNIST dataset (second row), the path of SGD satisfies iterationwise star-convexity for most of the iterations, except in the last several epochs where the training loss (see the red curve) already well saturates at zero value. In fact, the convergence is typically observed well before such a point. This is because when the training loss is very close to the global minimum (i.e., the gradient is very close to zero), small perturbation of the landscape easily deviates the SGD path from the desired star-convexity. Hence, our experiments demonstrate that SGD follows the iterationwise star-convex path up to the convergence occurs. Furthermore, on the CIFAR10 dataset (first row of Figure 3), we observe a strong evidence for the iterationwise star-convex path of SGD after several epochs of the initial phase of training. This implies that the loss landscape on a more challenging dataset can be more complex.
+
+Our empirical findings support the validity of the iterationwise star-convex path of SGD in a major part of practical training processes. Therefore, our convergence guarantee developed in Theorem 3 for SGD well justifies its practical success.
+
+![](images/b9d30d0acc1162b6b9829896132d8278b9aff55087890619dc53da069cd3d924.jpg)  
+Figure 4: Iterationwise path on local minimum.
+
+We next conduct further experiments to demonstrate that SGD follows the iterationwise star-convex path likely only for successful trainings to zero loss value, where a shared global minimum among all individual loss functions is achieved. To verify such a thought, we train an MLP using SGD on the CIFAR10 dataset under various settings with the number of hidden neurons ranging from 16 to 256. The results are shown in Figure 4, from which we observe that the training loss (i.e., red curves) converges to a non-zero value when the number of hidden neurons is small, implying that the algorithm likely attains a sub-optimal point which is not a common global minimum shared by all individual loss functions. In such trainings, we observe that the corresponding SGD paths have much fewer iterations satisfying the iterationwise star-convexity compared to the successful training instances shown in Figure 3. Thus, such empirical findings partially suggest that iterationwise star-convex SGD path more likely occurs when SGD can find a common global minimum, e.g., training overparameterized networks to zero loss value.
+
+# 5 CONCLUSION
+
+In this paper, we propose an epochwise star-convex property of the optimization path of SGD, which we validate in various experiments. Based on such a property, we show that SGD approaches a global minimum at an epoch level. Then, we further examine the property at an iteration level, and empirically show that it is satisfied in a major part of training processes. As we prove theoretically, such a more refined property guarantees the convergence of SGD to a global minimum, and the algorithm enjoys a self-variance-reducing effect. We believe that our study sheds light on the success of SGD in training neural networks from both empirical aspect and theoretical aspect.
+
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+
+# Supplementary Materials
+
+A PROOF OF THEOREM 1
+
+Observe that the SGD update can be rewritten as the following optimization step
+
+$$
+x _ { k + 1 } = \underset { u \in \mathbb { R } ^ { d } } { \mathrm { a r g m i n } } \Big \{ \underset { \ b { \xi } _ { k } } { \ell } ( x _ { k } ) + \langle u - x _ { k } , \nabla \ell _ { \xi _ { k } } ( x _ { k } ) \rangle + \frac { 1 } { 2 \eta } \| u - x _ { k } \| ^ { 2 } \Big \} .
+$$
+
+Note that the function $f _ { \xi _ { k } } ( u )$ is linear, and we further obtain that for all $x ^ { \ast } \in \mathcal { X } _ { \xi _ { k } } ^ { \ast }$
+
+$$
+\begin{array} { r l } { \eta \big ( f _ { \xi _ { k } } ( x _ { k + 1 } ) - f _ { \xi _ { k } } ( x ^ { * } ) \big ) = \eta \langle \nabla \ell _ { \xi _ { k } } ( x _ { k } ) , x _ { k + 1 } - x ^ { * } \rangle } & { } \\ { \overset { ( i ) } { = } \langle x _ { k } - x _ { k + 1 } , x _ { k + 1 } - x ^ { * } \rangle } & { } \\ { } & { = \cfrac { 1 } { 2 } \Big ( \| x _ { k } - x ^ { * } \| ^ { 2 } - \| x _ { k + 1 } - x ^ { * } \| ^ { 2 } - \| x _ { k + 1 } - x _ { k } \| ^ { 2 } \Big ) , } \end{array}
+$$
+
+where (i) uses the update rule of SGD. Rearranging the above inequality yields that
+
+$$
+f _ { \xi _ { k } } ( x _ { k + 1 } ) \leq f _ { \xi _ { k } } ( x ^ { * } ) + \frac { 1 } { 2 \eta } \Big ( \| x _ { k } - x ^ { * } \| ^ { 2 } - \| x _ { k + 1 } - x ^ { * } \| ^ { 2 } - \| x _ { k + 1 } - x _ { k } \| ^ { 2 } \Big ) .
+$$
+
+On the other hand, by smoothness of the loss function, we obtain that
+
+$$
+\begin{array} { r l } & { \ell _ { \xi _ { k } } ( x _ { k + 1 } ) \leq \ell _ { \xi _ { k } } ( x _ { k } ) + \langle x _ { k + 1 } - x _ { k } , \nabla \ell _ { \xi _ { k } } ( x _ { k } ) \rangle + \displaystyle \frac { L } { 2 } \| x _ { k + 1 } - x _ { k } \| ^ { 2 } } \\ & { \qquad = f _ { \xi _ { k } } ( x _ { k + 1 } ) + \displaystyle \frac { L } { 2 } \| x _ { k + 1 } - x _ { k } \| ^ { 2 } } \\ & { \qquad \overset { ( i ) } { \leq } f _ { \xi _ { k } } ( x ^ { * } ) + \displaystyle \frac { 1 } { 2 \eta } ( \| x _ { k } - x ^ { * } \| ^ { 2 } - \| x _ { k + 1 } - x ^ { * } \| ^ { 2 } ) - ( \displaystyle \frac { 1 } { 2 \eta } - \displaystyle \frac { L } { 2 } ) \| x _ { k + 1 } - x _ { k } \| ^ { 2 } } \\ & { \qquad \overset { ( i i ) } { \leq } f _ { \xi _ { k } } ( x ^ { * } ) + \displaystyle \frac { 1 } { 2 \eta } ( \| x _ { k } - x ^ { * } \| ^ { 2 } - \| x _ { k + 1 } - x ^ { * } \| ^ { 2 } ) , } \end{array}
+$$
+
+where (i) follows from eq. (3) and (ii) is due to the choice of learning rate. Summing the above inequality over $k$ from $n B$ to $n ( B + 1 ) - 1$ yields that, for every $x ^ { * } \in \mathcal { X } ^ { * }$ in Definition 1,
+
+$$
+\begin{array} { r } { \| x _ { n ( B + 1 ) } - x ^ { * } \| ^ { 2 } \leq \| x _ { n B } - x ^ { * } \| ^ { 2 } - \displaystyle \sum _ { k = n B \atop n ( B + 1 ) - 1 } ^ { n ( B + 1 ) - 1 } 2 \eta \big ( \ell _ { \xi _ { k } } ( x _ { k + 1 } ) - f _ { \xi _ { k } } ( x ^ { * } ) \big ) } \\ { \leq \| x _ { n B } - x ^ { * } \| ^ { 2 } - \displaystyle \sum _ { k = n B } ^ { n ( B + 1 ) - 1 } 2 \eta \big ( \ell _ { \xi _ { k } } ( x _ { k + 1 } ) - \ell _ { \xi _ { k } } ( x ^ { * } ) \big ) , } \end{array}
+$$
+
+where the last inequality follows from the star-convex path of SGD in Definition 1. The desired result follows from the above inequality and the fact that $\ell _ { \xi _ { k } } ( x ^ { * } ) = \operatorname* { i n f } _ { \boldsymbol { u } \in \mathbb { R } ^ { d } } \ell _ { \xi _ { k } } ( \boldsymbol { u } )$ for all $x ^ { * } \in \mathcal { X } ^ { * }$ Moreover, we conclude that the sequence $\{ x _ { n B } \} _ { B }$ is bounded. By continuity of $\nabla \ell _ { i }$ for all $i =$ $1 , . . . , n$ and the update rule of SGD, we further conclude that the entire sequence $\{ \boldsymbol { x } _ { k } \} _ { k }$ is bounded.
+
+# B PROOF OF THEOREM 2
+
+We first collect some facts. Recall that ${ { \mathcal X } ^ { * } } = \cap _ { i = 1 } ^ { n } { { \mathcal X } _ { i } ^ { * } }$ is non-empty and bounded. Consider any fixed $t \in \{ 0 , \ldots , n - 1 \}$ and recall that $k = n B + t$ , $\xi _ { k } = \pi _ { B } ( t + 1 )$ . Then, one can check that the iterations $k$ with $\xi _ { k } = \dot { v } \in \{ 1 , 2 , . . . , n \}$ form the subsequence $\{ x _ { n B + \pi _ { B } ^ { - 1 } ( v ) - 1 } \} _ { B }$ .
+
+Next, we prove item 1. Fix any $t \in \{ 0 , \ldots , n - 1 \}$ and sum eq. (4) over $k$ from $n B + t$ to $n ( B + 1 ) + t - 1$ yields that, for every $x ^ { * } \in \mathcal { X } ^ { * }$ in Definition 1,
+
+$$
+\| x _ { n ( B + 1 ) + t } - x ^ { * } \| ^ { 2 } \leq \| x _ { n B + t } - x ^ { * } \| ^ { 2 } - \sum _ { k = n B + t } ^ { n ( B + 1 ) + t - 1 } 2 \eta \big ( \ell _ { \xi _ { k } } ( x _ { k + 1 } ) - f _ { \xi _ { k } } ( x ^ { * } ) \big ) .
+$$
+
+Further summing the above inequality over $B$ from 0 to $K$ and rearranging, we obtain that
+
+$$
+\begin{array} { r l } { | x _ { n } ( K + 1 ) + t - x ^ { * } | | ^ { 2 } \le \| x _ { t } - x ^ { * } \| ^ { 2 } - } & { \displaystyle \sum _ { k = n } ^ { n ( K + 1 ) - 1 } 2 \eta \big ( \ell _ { \xi _ { k } } ( x _ { k + 1 } ) - f _ { \xi _ { k } } ( x ^ { * } ) \big ) } \\ & { \qquad - \displaystyle \sum _ { k = t } ^ { n - 1 } 2 \eta \big ( \ell _ { \xi _ { k } } ( x _ { k + 1 } ) - f _ { \xi _ { k } } ( x ^ { * } ) \big ) - \displaystyle \sum _ { k = n ( K + 1 ) } ^ { n ( K + 1 ) + t - 1 } 2 \eta \big ( \ell _ { \xi _ { k } } ( x _ { k + 1 } ) - f _ { \xi _ { k } } ( x ^ { * } ) \big ) , } \\ { \le \| x _ { t } - x ^ { * } \| ^ { 2 } - } & { \displaystyle \sum _ { k = n } ^ { n ( K + 1 ) - 1 } 2 \eta \big ( \ell _ { \xi _ { k } } ( x _ { k + 1 } ) - \ell _ { \xi _ { k } } ( x ^ { * } ) \big ) } \\ & { \qquad - \displaystyle \sum _ { k = t } ^ { n - 1 } 2 \eta \big ( \ell _ { \xi _ { k } } ( x _ { k + 1 } ) - f _ { \xi _ { k } } ( x ^ { * } ) \big ) - \displaystyle \sum _ { k = n ( K + 1 ) } ^ { n ( K + 1 ) + t - 1 } 2 \eta \big ( \ell _ { \xi _ { k } } ( x _ { k + 1 } ) - f _ { \xi _ { k } } ( x ^ { * } ) \big ) , } \end{array}
+$$
+
+where the last inequality follows from the the star-convex path of SGD in Definition 1. Consider the term $\ell _ { \xi _ { k } } ( x _ { k + 1 } ) - \ell _ { \xi _ { k } } ( x ^ { * } )$ in eq. (6) along the iterations with $\xi _ { k } = v \in \{ 1 , 2 , . . . , n \}$ . Such term can be rewritten as $\ell _ { v } ( \bar { x } _ { n B + \pi _ { B } ^ { - 1 } ( v ) } ) - \ell _ { v } ( x ^ { * } )$ . Suppose for certain $v \in \{ 1 , 2 , . . . , n \}$ the sequence $\{ \ell _ { v } ( x _ { n B + \pi _ { B } ^ { - 1 } ( v ) } ) \} _ { B }$ does not converge to its global minimum $\operatorname* { i n f } _ { \substack { x \in \mathbb { R } ^ { d } } } \ell _ { v } ( x )$ . Then, by the cyclic sampling scheme with reshuffle, we conclude that the first summation term in eq. (6) diverges to $+ \infty$ as $K  \infty$ . Also, note that the last two summation terms have finite number of elements, which are all bounded as $\{ \boldsymbol { x } _ { k } \} _ { k }$ is bounded. Therefore, we conclude that the sequences $\{ x _ { n B + t } \} _ { B }$ for $t = 0 , . . . , n - 1$ converge to $x ^ { * }$ for all candidates $x ^ { * } \in \mathcal { X } ^ { * }$ in Definition 1. Next, consider the case in which there are multiple such candidate $x ^ { * } s$ in Definition 1. Then, the previous sentence states that $\{ x _ { n B + t } \} _ { B }$ converges to multiple limits, which cannot happen for a convergent sequence. This leads to a contradiction. Consider the other case that there is only one such candidate $x ^ { * }$ in Definition 1. Then, we conclude that all the sequences $\{ x _ { n B + t } \} _ { B }$ for $t = 0 , . . . , n - 1$ converge to $x ^ { * }$ , i.e., the entire sequence $\{ x _ { k } \} _ { k }$ converges to such unique common global minimizer. This contradicts with our assumption that $\{ \ell _ { v } ( x _ { n B + \pi _ { B } ^ { - 1 } ( v ) } ) \} _ { B }$ does not converge to $\operatorname* { i n f } _ { \substack { x \in \mathbb { R } ^ { d } } } \ell _ { v } ( x )$ for certain $v$ . Combining both cases, we obtain the desired claim of item 1.
+
+Next, we prove item 2. Note that sequence $\{ x _ { k } \} _ { k }$ is bounded . Fix any $v \in \{ 1 , \ldots , n \}$ and consider any limit point $z _ { v }$ of $\{ x _ { n B + \pi _ { B } ^ { - 1 } ( v ) } \} _ { B }$ , i.e., $x _ { n B _ { j } + \pi _ { B _ { j } } ^ { - 1 } ( v ) } \stackrel { j } { \to } z _ { v }$ along a proper subsequence. From item 1 we know that $\begin{array} { r } { \ell _ { v } \big ( x _ { n B _ { j } + \pi _ { B _ { j } } ^ { - 1 } ( v ) } \big ) \stackrel { j } { \to } \operatorname* { i n f } _ { x \in \mathbb { R } ^ { d } } \ell _ { v } ( x ) } \end{array}$ . This, together with the continuity of the loss function, implies that $z _ { v } \in \mathcal { X } _ { v } ^ { * }$ for all $v$ .
+
+# C PROOF OF THEOREM 3
+
+Recall that eq. (4) shows that, for all $x ^ { \ast } \in \mathcal { X } _ { \xi _ { k } } ^ { \ast }$
+
+$$
+\begin{array} { r l r } & { } & { \ell _ { \xi _ { k } } ( x _ { k + 1 } ) \le f _ { \xi _ { k } } ( x ^ { * } ) + \displaystyle \frac { 1 } { 2 \eta } \Big ( \| x _ { k } - x ^ { * } \| ^ { 2 } - \| x _ { k + 1 } - x ^ { * } \| ^ { 2 } \Big ) } \\ & { } & { \stackrel { ( i ) } { \le } \ell _ { \xi _ { k } } ( x ^ { * } ) + \displaystyle \frac { 1 } { 2 \eta } \Big ( \| x _ { k } - x ^ { * } \| ^ { 2 } - \| x _ { k + 1 } - x ^ { * } \| ^ { 2 } \Big ) , } \end{array}
+$$
+
+where (i) follows from the iterationwise star-convex path in Definition 2. Since $\ell _ { \xi _ { k } } ( x _ { k + 1 } ) - \ell _ { \xi _ { k } } ( x ^ { * } ) \geq$ 0, we conclude that for all $k = 0 , 1 , \ldots$ and every $x ^ { \ast } \in \mathcal { X } _ { \xi _ { k } } ^ { \ast }$ ,
+
+$$
+\lVert x _ { k + 1 } - x ^ { * } \rVert \leq \lVert x _ { k } - x ^ { * } \rVert .
+$$
+
+Next, consider any $v \in \{ 1 , . . . , n \}$ , we show that every limit point $z _ { v }$ of $\{ x _ { n B + \pi _ { B } ^ { - 1 } ( v ) } \} _ { B }$ is in $\mathcal { X } ^ { \ast }$ . By eq. (9), we know that $\{ \| x _ { n B + \pi _ { B } ^ { - 1 } ( v ) } - x ^ { * } \| \} _ { B }$ is decreasing. Consider a limit point $z _ { v }$ associated with the subsequence such that $x _ { n B _ { j } + \pi _ { B _ { j } } ^ { - 1 } ( v ) } \to z _ { v }$ . Then, for all $B _ { j } \geq B$ we know that, for any fixed $x ^ { * } \in \mathcal { X } ^ { * }$ ,
+
+$$
+\| z _ { v } - x ^ { * } \| \overset { j } {  } \| x _ { n B _ { j } + \pi _ { B _ { j } } ^ { - 1 } ( v ) } - x ^ { * } \| \leq \| x _ { n B + \pi _ { B } ^ { - 1 } ( v ) } - x ^ { * } \| .
+$$
+
+Next, we prove by contradiction. Suppose that $z _ { v } \notin \mathcal { X } ^ { \ast }$ . Then, eq. (10) implies that $\parallel x _ { n B + \pi _ { B } ^ { - 1 } ( v ) } -$ $x ^ { * } \Vert > 0$ for all large $B$ and any $x ^ { \ast } \in \mathcal { X } ^ { \ast }$ . Combining this conclusion with item 2 of Theorem 2, it follows that all the limit points of $\{ x _ { n B + \pi _ { B } ^ { - 1 } ( v ) } \} _ { B }$ are in ${ \mathcal { X } } _ { v } ^ { * } \mid { \mathcal { X } } ^ { * }$ . Since ${ { \mathcal X } ^ { * } } = \cap _ { i = 1 } ^ { n } { { \mathcal X } _ { i } ^ { * } }$ , it follows that the limit points of $\{ x _ { n B + \pi _ { B } ^ { - 1 } ( u ) } \} _ { B }$ are different from those of $\{ x _ { n B + \pi _ { B } ^ { - 1 } ( v ) } \} _ { B }$ for any $u \ne v$ . Now consider a subsequence $x _ { n B _ { j } + \pi _ { B _ { j } } ^ { - 1 } ( v ) } \to z _ { v } \in \mathcal { X } _ { v } ^ { * } \setminus \mathcal { X } ^ { * }$ . Also, consider the subsequence $\{ x _ { n B _ { j } + \pi _ { B _ { j } } ^ { - 1 } ( u ) } \} _ { j }$ with bseq $u \ne v$ he  of lic samplisuch that hat with probability(this occurs with a $B _ { j ( s ) }$ $B _ { j }$ $\pi _ { B _ { j ( s ) } } ^ { - 1 } ( u ) = \pi _ { B _ { j ( s ) } } ^ { - 1 } ( v ) + 1$ constant probability in every epoch). Applying eq. (9) along this subsequence, we conclude that
+
+$$
+\begin{array} { r } { \| x _ { n B _ { j ( s ) } + \pi _ { B _ { j ( s ) } } ^ { - 1 } ( u ) } - z _ { v } \| \leq \| x _ { n B _ { j ( s ) } + \pi _ { B _ { j ( s ) } } ^ { - 1 } ( v ) } - z _ { v } \| \overset { j } {  } 0 . } \end{array}
+$$
+
+Let j → ∞ in the above equation, we conclude that xnBj(s)+π−1Bj(s)(u ) → zv , i.e., zv is a limit point of $\{ x _ { n B _ { j } + \pi _ { B _ { j } } ^ { - 1 } ( u ) } \} _ { B }$ . Note that $z _ { v }$ is a limit point of $\{ x _ { n B + \pi _ { B } ^ { - 1 } ( v ) } \} _ { B }$ . This contradicts our previous conclusion that the limit points of $\{ x _ { n B + \pi _ { B } ^ { - 1 } ( u ) } \} _ { B }$ must be different from those of $\{ x _ { n B + \pi _ { B } ^ { - 1 } ( v ) } \} _ { B }$ for $u \ne v$ . Thus, we must have for all $v = 1 , \ldots , n$ , every limit point $z _ { v }$ of $\{ x _ { n B + \pi _ { B } ^ { - 1 } ( v ) } \} _ { B }$ is in $\mathcal { X } ^ { \ast }$ Then, by eq. (9) we further conclude that for all $B = 0 , 1 , . . .$ .
+
+$$
+\| x _ { n B + \pi _ { B } ^ { - 1 } ( v ) } - z _ { v } \| \leq \| x _ { n ( B - 1 ) + \pi _ { B - 1 } ^ { - 1 } ( v ) } - z _ { v } \| .
+$$
+
+Note that $x _ { n B _ { j } + \pi _ { B _ { j } } ^ { - 1 } ( v ) } \overset { j } { \to } z _ { v }$ . Thus, for all $B \geq B _ { j }$ the above inequality implies that
+
+$$
+\begin{array} { r } { \| x _ { n B + \pi _ { B } ^ { - 1 } ( v ) } - z _ { v } \| \le \| x _ { n B _ { j } + \pi _ { B _ { j } } ^ { - 1 } ( v ) } - z _ { v } \| \overset { j } {  } 0 . } \end{array}
+$$
+
+This shows that $\{ x _ { n B + \pi _ { B } ^ { - 1 } ( v ) } \} _ { B }$ has a unique limit point $z _ { v }$ , which is an element of $\mathcal { X } ^ { \ast }$ . Next, consider the limits $z _ { u } , z _ { v } ( u \ne v )$ of the sequences $\{ x _ { n B + \pi _ { B } ^ { - 1 } ( u ) } \} _ { B } , \{ x _ { n B + \pi _ { B } ^ { - 1 } ( v ) } \} _ { B }$ , respectively. By eq. (9) and the fact that $z _ { u } \in \mathcal { X } ^ { * }$ , we conclude that
+
+$$
+\begin{array} { r } { \| x _ { n B + \pi _ { B } ^ { - 1 } ( v ) } - z _ { u } \| \le \| x _ { n ( B - 1 ) + \pi _ { B - 1 } ^ { - 1 } ( v ) } - z _ { u } \| \le \| x _ { n ( B - 2 ) + \pi _ { B - 2 } ^ { - 1 } ( u ) } - z _ { u } \| \overset { B } { \to } 0 . } \end{array}
+$$
+
+Thus, $x _ { n B + \pi _ { B } ^ { - 1 } ( v ) }  z _ { u }$ , and we conclude that $z _ { v } = z _ { u }$ for all $v \neq u$ , i.e., the whole sequence $\{ x _ { k } \}$ has a unique limit point in $\mathcal { X } ^ { \ast }$ .
+
+# D SUPPLEMENTARY EXPERIMENTS
+
+In this section, we provide more experiments to illustrate the star-convexity property of the SGD path from other different aspects.
+
+# Verification of epochwise star-convexity with different reference points
+
+In Figure 5 we verify the epochwise star-convexity of the SGD path by setting the reference point $x ^ { * }$ to be the output of SGD at different intermediate epochs (i.e., 60,80,100,120 epochs), where the SGD has already saturated close to zero loss. The experiments are conducted by training the Alexnet and MLP on Cifar10 using SGD. As can be seen from Figure 5, the epochwise star-convexity still hold (i.e., $e _ { B } < 0$ ) after certain epochs in the initial training phase. This shows that the observed star-convex path does not depend on the choice of reference point (so long as they achieve near-zero loss).
+
+![](images/47b6aa52170926d76249d7e2630f5047ce2998413c192414bcaf31d053cf811c.jpg)  
+Figure 5: Verification of epochwise star-convexity when the reference point $x ^ { * }$ is taken at end of different epochs. Top plots correspond to the training of Alexnet with $x ^ { * }$ taken at 60th, 80th, 100th, and 120th epochs, and bottom plots correspond to the training of MLP with $x ^ { * }$ taken at 60th, 80th, $1 0 0 \mathrm { { t h } }$ , and $1 2 0 \mathrm { { t h } }$ epochs.
+
+# Growth of weight norm in neural network training
+
+In Figure 6, we present the growth of the $\ell _ { 2 }$ norm of network weights in training different neural networks on Cifar10 under the cross-entropy loss. It can be seen from the figure that the norm of the weights increases slowly (logarithmly) after the training loss achieves near-zero. This is because the gradient is nearly zero when the training is close to the global minimum, and therefore the updates of the weights are very small.
+
+![](images/1d3f0254a74c42f2b28e56e6beee460c762a4e53e11a64623288eb09a4b60294.jpg)  
+Figure 6: $\ell _ { 2 }$ norm of network weights and training loss of different networks.
+
+# Verification of star-convexity under MSE loss
+
+In Figure 7, we verify the epochwise star-convexity by training different networks on MNIST dataset under the MSE loss (i.e., $\ell _ { 2 }$ loss). We note that unlike the cross-entropy loss, zero value can be achieved by the MSE loss. We set the reference point $x ^ { * }$ to be the output of SGD at the 40th epoch. It can be seen from the figure that the residue $e _ { B }$ is negative along the entire optimization path, demonstrating that the SGD path satisfies the epochwise star-convexity under the MSE loss.
+
+![](images/6aba459d7f1438c7676dbf6225b82fff306f10712555be2b4ebc437c352d2b13.jpg)  
+Figure 7: Verification of epochwise star-convexity under MSE loss (i.e., $\ell _ { 2 }$ loss).

parse/train/BylIciRcYQ/BylIciRcYQ_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "SGD CONVERGES TO GLOBAL MINIMUM IN DEEP LEARNING VIA STAR-CONVEX PATH ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            820,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yi Zhou∗, Junjie Yang†, Huishuai Zhang‡, Yingbin Liang§, Vahid Tarokh∗ ∗Duke University, †University of Science and Technology of China ‡Microsoft Research, Asia, §The Ohio State University ",
+        "bbox": [
+            184,
+            167,
+            700,
+            213
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            251,
+            544,
+            266
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Stochastic gradient descent (SGD) has been found to be surprisingly effective in training a variety of deep neural networks. However, there is still a lack of understanding on how and why SGD can train these complex networks towards a global minimum. In this study, we establish the convergence of SGD to a global minimum for nonconvex optimization problems that are commonly encountered in neural network training. Our argument exploits the following two important properties: 1) the training loss can achieve zero value (approximately), which has been widely observed in deep learning; 2) SGD follows a star-convex path, which is verified by various experiments in this paper. In such a context, our analysis shows that SGD, although has long been considered as a randomized algorithm, converges in an intrinsically deterministic manner to a global minimum. ",
+        "bbox": [
+            233,
+            285,
+            764,
+            438
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            472,
+            336,
+            488
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Training neural networks has been proven to be NP-hard decades ago Blum & Rivest (1988). At that time, the limited computation power makes neural network training a “mission impossible”. However, as the development of computing device achieves several revolutionary milestones (e.g. GPUs), deep neural networks are found to be practically trainable and can generalize well on real datasets Krizhevsky et al. (2017). At the same time, deep learning technique starts to beat the performance of other conventional approaches in a variety of challenging tasks, e.g., computer vision, classification, natural language processing, etc. ",
+        "bbox": [
+            174,
+            507,
+            826,
+            604
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Modern neural network training is typically performed by applying first-order algorithms such as stochastic gradient descent (SGD) (a.k.a backpropagation) (Linnainmaa, 1976) or its variants, e.g., Adam (Kingma & Ba, 2014), Adagrad (Duchi et al., 2011), etc. Traditional analysis of SGD in nonconvex optimization guarantees the convergence to a stationary point Bottou et al. (2016); Ghadimi et al. (2016). Recently, it has been shown that SGD has the capability to escape strict saddle points Ge et al. (2015); Jin et al. (2017); Reddi et al. (2018a); Daneshmand et al. (2018), and can escape even sharp local minima Kleinberg et al. (2018). While these works provide different insights towards understanding the performance of SGD, they cannot explain the success of SGD that has been widely observed in deep learning applications. Specifically, it is known that SGD is able to train a variety of deep neural networks to achieve zero training loss (either exactly or approximately) for non-negative loss functions. This implies that SGD can find a global minimum of deep neural networks at ease. The major challenges towards understanding this phenomenon are in two-fold: 1) deep neural networks have complex landscape that cannot be fully understood analytically (Zhou & Liang, 2018); 2) the randomness nature of SGD makes it hard to characterize its convergence on such a complex landscape. These factors prohibit a good understanding of the practical success of SGD in deep learning from a traditional optimization aspect. ",
+        "bbox": [
+            174,
+            611,
+            825,
+            833
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this study, we analyze the convergence of SGD in the above phenomenon by exploiting the following two critical properties. First, the fact that SGD can train neural networks to zero loss value implies that the non-negative loss functions on all data samples share a common global minimum. Second, our experiments establish strong empirical evidences that SGD (when training the loss to zero value) follows a star-convex path. Based on these properties, we formally establish the convergence of SGD to a global minimizer. Our work conveys a useful insight that although the landscape of neural networks can be complicated, the actual optimization path that SGD takes turns out to be remarkably simple and sufficient to guarantee the converge to a global minimum. ",
+        "bbox": [
+            174,
+            840,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1.1 OUR CONTRIBUTIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            150,
+            369,
+            165
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We focus on the empirical observation that SGD can train various neural networks to achieve zero loss, and validate that SGD follows an epochwise star-convex path in empirical optimization processes. Based on such a property, we prove that the Euclidean distance between the variable sequence generated by SGD and a global minimizer decreases at an epoch level. We also show that the subsequences of iterations that correspond to the same data sample is a minimizing sequence of the loss corresponding to that sample. ",
+        "bbox": [
+            174,
+            176,
+            825,
+            261
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "By further empirical exploration, we validate that SGD follows an iterationwise star-convex path during the major part of the training process. Based on such a property, we prove that the entire variable sequence generated by SGD converges to a global minimizer of the objective function. Then, we show that the convergence of SGD induces a self-regularization on its variance, i.e., the variance of stochastic gradients vanishes as SGD converges in such a context. ",
+        "bbox": [
+            174,
+            267,
+            825,
+            337
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "From a technical perspective, we characterize the intrinsic deterministic convergence property of SGD when the optimization path is well regularized, rather than the performance on average or in probability established in the existing studies. Our results provide a novel and promising aspect to understand SGD-based optimization in deep learning. Furthermore, our analysis of SGD explores the limiting convergence property of the subsequences that correspond to individual data samples, which is in sharp contrast to the traditional treatment of SGD that depends on its random nature and bounds on variance. Hence, our proof technique can be of independent interest to the community. ",
+        "bbox": [
+            174,
+            344,
+            825,
+            441
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1.2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            459,
+            330,
+            474
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "As there are extensive literature on SGD, we only mention the highly relevant studies here. Theoretical foundations of SGD have been developed in the optimization community (Robbins & Monro, 1951; Nemirovski et al., 2009; Lan, 2012; Ghadimi et al., 2016; Ghadimi & Lan, 2016), and have attracted much attention from the machine learning community in the past decade (Schmidt et al., 2017; Defazio et al., 2014; Johnson & Zhang, 2013; Li et al., 2017; Wang et al., 2018a). In general nonconvex optimization, it is known that SGD converges to a stationary point under a bounded variance assumption and a diminishing learning rate Bottou et al. (2016); Ghadimi et al. (2016). Other variants of SGD that are designed for deep learning have been proposed, e.g., Adam (Kingma & Ba, 2014), AMSgrad Reddi et al. (2018b), Adagrad (Duchi et al., 2011), etc, and their convergence properties have been studied in the context of online convex regret minimization. ",
+        "bbox": [
+            174,
+            486,
+            825,
+            626
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Needell et al. (2014); Moulines & Bach (2011) show that SGD converges at a linear rate when the objective function is strongly convex and has a unique common global minimum. Recently, several studies show that SGD has the capability to escape strict saddle points Ge et al. (2015); Jin et al. (2017); Reddi et al. (2018a); Daneshmand et al. (2018). Other cubic-regularization-based methods have also been shown to be able to escape strict saddle points Nesterov & Polyak (2006); Zhou et al. (2018); Wang et al. (2018b). Kleinberg et al. (2018) considers functions with one-point strong convexity, and shows that the randomness of SGD has an intrinsic smoothing effect that can avoid convergence to sharp minimum. Our paper exploits a very different notion of star-convexity path of SGD, which is a much weaker condition than those in the previous studies. ",
+        "bbox": [
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+            757
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 PROBLEM SETUP AND PRELIMINARIES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            779,
+            529,
+            795
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Neural network training can be formulated as the following finite-sum optimization problem. ",
+        "bbox": [
+            169,
+            810,
+            781,
+            827
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/809a331a140635346db93d76001f3a999f44759674d5dd1cabd18d78e7260de2.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { x \\in \\mathbb { R } ^ { d } } f ( x ) : = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\ell _ { i } ( x ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            410,
+            833,
+            588,
+            875
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where there are in total $n$ training data samples. The loss function that corresponds to the $i$ -th data sample is denoted by $\\ell _ { i } : \\mathbb { R } ^ { \\breve { d } } \\to \\mathbb { R }$ for $i = 1 , \\ldots , n$ , and the vector to be minimized in the problem (e.g., network weights in deep learning) are denoted by $x \\in \\mathbb { R } ^ { d }$ . In general, problem $( \\mathrm { P } )$ is a nonconvex optimization problem, and we make the following standard assumptions regarding the objective function. ",
+        "bbox": [
+            174,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Assumption 1. The loss functions $\\ell _ { i } , i = 1 , \\ldots , n$ in problem $( P )$ satisfy: ",
+        "bbox": [
+            176,
+            136,
+            658,
+            151
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "1. They are continuously differentiable, and their gradients are $L$ -Lipschitz continuous; ",
+        "bbox": [
+            174,
+            161,
+            746,
+            178
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2. For every $i = 1 , \\dots , n , \\operatorname* { i n f } _ { x \\in \\mathbb { R } ^ { d } } \\ell _ { i } ( x ) > 0 .$ ",
+        "bbox": [
+            176,
+            185,
+            473,
+            202
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The conditions imposed by Assumption 1 are standard in analysis of nonconvex optimization. In specific, item 1 is a standard smoothness assumption on nonconvex loss functions. Item 2 assumes that the loss functions are bounded below, which is satisfied by many loss functions in deep learning, e.g., MSE loss, crossentropy loss, NLL loss, etc, which are all non-negative. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Next, we introduce the following fact that is widely observed in training overparameterized deep neural networks, which we assume to hold throughout the paper. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Observation 1 (Global minimum in deep learning). The objective function $f$ in problem $( P )$ with non-negative loss can achieve zero value at certain $x ^ { * }$ . Thus, $x ^ { * }$ is also a common global minimizer for all individual loss $\\{ \\ell _ { i } \\} _ { i = 1 } ^ { n }$ . More formally, denote $\\mathcal { X } _ { i } ^ { \\ast }$ as the set of global minimizers of $\\ell _ { i }$ for $i = 1 , \\ldots , n$ . Then, the set of common global minimizers, i.e., $\\mathcal { X } ^ { * } : = \\cap _ { i = 1 } ^ { n } \\mathcal { X } _ { i } ^ { * }$ , is non-empty and bounded. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Observation 1 is a common observation in deep learning applications, because deep neural networks (especially in the overparameterized regime) typically have enough capacity to fit all training data samples, and therefore the model has a global minimum shared by the loss functions on all data samples. To elaborate more, if the total loss $f$ attains zero value at $x ^ { * }$ , then $\\ell _ { i }$ for all $i$ must achieve zero value at $x ^ { * }$ as they are non-negative. Thus, $x ^ { * }$ is a common global minimum of all individual loss $\\ell _ { i } ( x )$ for all $i$ . Such a fact plays a critical role in understanding the convergence of SGD in training neural networks. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Next, we introduce our algorithm of interest – stochastic gradient descent (SGD). Specifically, to solve problem (P), SGD starts at an initial vector $x _ { 0 } \\in \\mathbb { R } ^ { d }$ and generates a variable sequence $\\{ x _ { k } \\} _ { k }$ according to the following update rule. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ac34cf3a28ff9248ebf60061426fd1a56190a2d2ce09767be9b4cc34e1e2c3ca.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\mathrm { ( S G D ) } } & { { } x _ { k + 1 } = x _ { k } - \\eta \\nabla \\ell _ { \\xi _ { k } } ( x _ { k } ) , \\quad k = 0 , 1 , . . . , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\eta > 0$ denotes the learning rate, and $\\xi _ { k } \\in \\{ 1 , \\ldots , n \\}$ corresponds to the sampled data index at iteration $k$ . In this study, we consider the practical cyclic sampling scheme with reshuffle (also referred to as the random sampling without replacement) to generate the random variable $\\xi _ { k }$ . To elaborate the notation, we rewrite every iteration number $k$ as $n B + t$ , where $B = 0 , 1 , 2 , \\ldots$ denotes the index of epoch that iteration $k$ belongs to and $t \\in \\{ 0 , 1 , . . . , n - 1 \\}$ denotes the corresponding iteration index in that epoch. We further denote $\\pi _ { B }$ as the random permutation of $1 , . . . , n$ in the $B$ -th epoch, and denote $\\pi _ { B } ( j )$ as its $j$ -th element. Then, the sampled data index at iteration $k$ can be expressed as ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a62068d570744049905027d31e0294c948fe049921c8fed25265aebe4e38b3ec.jpg",
+        "text": "$$\n\\xi _ { k } = \\pi _ { B } ( t + 1 ) , \\quad t = 0 , . . . , n - 1 .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 APPROACHING GLOBAL MINIMUM EPOCHWISELY ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 AN INTERESTING EMPIRICAL OBSERVATION OF SGD PATH ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            751,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we provide empirical observations on the algorithm path of SGD in training neural networks. To be specific, we train a standard multi-layer perceptron (MLP) network Krizhevsky (2009), a variant of Alexnet and a variant of Inception network Zhang et al. (2017a) on the CIFAR10 dataset Krizhevsky (2009) using SGD under crossentropy loss. In all experiments, we adopt a constant learning rate (0.01 for MLP and Alexnet, 0.1 for Inception) and a constant mini-batch size 128. We discard all other optimization features such as momentum, weight decay, dropout and batch normalization, etc, in order to observe the essential property of SGD. ",
+        "bbox": [
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+            776,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In each experiment, we train the network for a sufficient number of epochs to achieve near-zero training loss (i.e., almost global minimum), and record the weight parameters along the iteration path of SGD. We denote the weight parameters produced by SGD in the last iteration as $x ^ { * }$ , which has a near zero loss, and evaluate the Euclidean distance between the weight parameters produced by SGD and $x ^ { * }$ along the iteration path. We plot the results in Figure 1. It can be seen that the training losses for all three networks fluctuate along the iteration path, implying that the algorithm passes through complex landscapes. However, the Euclidean distance between the weight parameters and the final output $x ^ { * }$ is monotonically decreasing epochwise along the SGD path for all three networks. This shows that the variable sequence generated by SGD approaches the global minimum $x ^ { * }$ in a remarkably stable way. This motivates us to explore the underlying mechanism that yields such interesting observations. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In the next two subsections, we first propose a property that the algorithm path of SGD satisfies, based on which we formally prove that the variable sequence generated by SGD admits the behavior observed in Figure 1. Then, we provide empirical evidences to validate such a property of SGD path in practical SGD training. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/c122fe5d26fdcae1410b98a7230ad63527b055c7fd9264d47dc9bc4cb09ff6a1.jpg",
+        "image_caption": [
+            "Figure 1: Distance to output of SGD in training neural networks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 EPOCHWISE STAR-CONVEX PATH ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we introduce the notion of the epochwise star-convex path for SGD and establish its theoretical implications on the convergence of SGD. We validate that SGD satisfies such a property in practical neural network training in Section 3.3. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Recall the conventional definition of star-convexity. Let $x ^ { * }$ be a global minimizer of a smooth function $h$ . Then, $h$ is said to be star-convex at a point $x$ provided that ",
+        "bbox": [
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+            545,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2ed63646230f2823b7603eb17ccb6aefa07f9bd741c5f4ee6a8c97aa83f25872.jpg",
+        "text": "$$\n( { \\mathrm { S t a r - c o n v e x i t y } } ) : \\quad h ( x ) - h ( x ^ { * } ) + \\langle x ^ { * } - x , \\nabla h ( x ) \\rangle \\leq 0 .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Star-convexity can be intuitively understood as convexity between a reference point $x$ and a global minimizer $x ^ { * }$ . Such a property ensures that the negative gradient $- \\nabla h ( x )$ points to the desired direction $x ^ { * } - x$ for minimization. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Next, we define the notion of epochwise star-convex path, which requires the star-convexity to be held cumulatively over each epoch. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 1 (Epochwise star-convex path). We call a path generated by SGD epochwise star-convex if it satisfies: For all epochs $B = 0 , 1 , \\ldots$ and for a fixed $x ^ { \\ast } \\in \\mathcal { X } ^ { \\ast }$ (see Observation 1 for definition), ",
+        "bbox": [
+            173,
+            679,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cc76f54706296f6da0f5dee556002e27844966ca0e9ee486cef10fa659cde80d.jpg",
+        "text": "$$\n\\sum _ { k = n B } ^ { n ( B + 1 ) - 1 } \\left[ \\ell _ { \\xi _ { k } } ( x _ { k } ) - \\ell _ { \\xi _ { k } } ( x ^ { * } ) + \\langle x ^ { * } - x _ { k } , \\nabla \\ell _ { \\xi _ { k } } ( x _ { k } ) \\rangle \\right] \\leq 0 .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We note that the property introduced by Definition 1 is not about the landscape geometry of a loss function, which can be complex as observed in the training loss curves shown in Figure 1. Rather, it characterizes the interaction between the algorithm and the loss function along the optimization path. Such a property is generally weaker than the global star-convexity, and is observed to be held in practical neural network training (see Section 3.3). ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Based on Definition 1, we obtain the following property of SGD. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1 (Epochwise diminishing distance). Let Assumption 1 hold and apply SGD with learning rate $\\begin{array} { r } { \\eta < { \\frac { 1 } { L } } } \\end{array}$ to solve problem $( P )$ . Assume SGD follows an epochwise star-convex path for a certain $x ^ { * } \\in \\mathcal { X } ^ { * }$ . Then, the variable sequence $\\{ x _ { k } \\} _ { k }$ generated by SGD satisfies, for all epochs $B = 0 , 1 , . . . ,$ ",
+        "bbox": [
+            174,
+            862,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/77b62a83e045e5451ef237c80498332f7917b903ae676f8f6ad91912c63c92e7.jpg",
+        "text": "$$\n\\| x _ { n ( B + 1 ) } - x ^ { * } \\| \\leq \\| x _ { n B } - x ^ { * } \\| .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            388,
+            909,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1 proves that the variable sequence generated by SGD approaches a global minimizer at an epoch level, which is consistent with the empirical observations made in Figure 1. Therefore, the property of epochwise star-convex path of SGD is sufficient to explain such desirable empirical observations, although the loss function can be highly nonconvex and has complex landscape. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            160
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Under the cyclic sampling scheme with reshuffle, SGD samples every data sample once per epoch. Consider the loss $\\ell _ { v }$ on the $v$ -th data sample for a fixed $v \\in \\{ 1 , 2 , . . . , n \\}$ . One can check that the iterations in which the loss $\\ell _ { v }$ is sampled form a subsequence $\\{ x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } \\} _ { B }$ , where $\\pi _ { B } ^ { - 1 }$ is the inverse permutation mapping of $\\pi _ { B }$ , i.e., $\\pi _ { B } ^ { - 1 } ( u ) = v$ if and only if $\\pi _ { B } ( v ) = u$ . Next, we characterize the convergence properties of these subsequences corresponding to the loss functions on individual data samples. ",
+        "bbox": [
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+            166,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2 (Minimizing subsequences). Under the same settings as those of Theorem 1, the subsequences $\\{ x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } \\} _ { B }$ for $v = 1 , . . . , n$ satisfy ",
+        "bbox": [
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+            261,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "1. They are minimizing sequences for the corresponding loss functions, i.e., ",
+        "bbox": [
+            176,
+            304,
+            668,
+            320
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/07189cbfce314ada85aa5eb3b6ad357ddb20279b9703d4cacd3aef6dda4d8485.jpg",
+        "text": "$$\n\\operatorname* { l i m } _ { B \\to \\infty } \\ell _ { v } ( x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } ) = \\operatorname* { i n f } _ { x \\in \\mathbb { R } ^ { d } } \\ell _ { v } ( x ) , \\quad \\forall v \\in \\{ 1 , . . . , n \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2. Every limit point of $\\{ x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } \\} _ { B }$ is in $\\mathcal { X } _ { v } ^ { \\ast }$ ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2 characterizes the limiting behavior of the subsequences that correspond to the loss functions on individual data samples. Essentially, the results in items 1 and 2 show that each subsequence $\\{ x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } \\} _ { B }$ is a minimizing sequence for the corresponding loss $\\ell _ { v }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            457,
+            263,
+            472
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We note that Theorems 1 and 2 characterize the epochwise convergence property of SGD in a deterministic way. The underlying technical reason is that the common global minimizer structure in Observation 1 suppresses the randomness induced by sampling and reshuffling of SGD, and ensures a common direction along which SGD can approach the global minimum on all individual data samples. Such a result is very different from traditional understanding of SGD where randomness and variance play a central role Nemirovski et al. (2009); Ghadimi et al. (2016). ",
+        "bbox": [
+            174,
+            483,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 VERIFYING EPOCHWISE STAR-CONVEX PATH OF SGD ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            583,
+            591,
+            598
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we conduct experiments to validate the epochwise star-convex path of SGD introduced in Definition 1. ",
+        "bbox": [
+            176,
+            609,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We train the aforementioned three types of neural networks, i.e., MLP, Alexnet and Inception, on CIFAR10 Krizhevsky (2009) and MNIST Lecun et al. (1998) dataset using SGD. The hyperparameter settings are the same as those mentioned in Section 3.1. We train these networks for a sufficient number of epochs to achieve a near-zero training loss (i.e., near-global minimum). We record the variable sequence generated by SGD along the entire algorithm path, and set $x ^ { * }$ to be the final output of SGD. Then, we evaluate the value of the summation term in Definition 1 for each epoch. The value of this summation term for each epoch $B$ is denoted as residual $e _ { B }$ . By Definition 1, SGD path in the $B$ -th epoch is epochwise star-convex provided that $e _ { B } < 0$ . ",
+        "bbox": [
+            174,
+            645,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Figure 2 shows the results of our experiments. In all subfigures, the red horizontal curve denotes the zero value baseline, and the other curve denotes the residual $e _ { B }$ . It can be seen from Figure 2 that, on the MNIST dataset (second row), the entire path of SGD satisfies epochwise star-convexity for all three networks. On the CIFAR10 dataset (first row), we observe an epochwise star-convex path of SGD after several epochs of the initial phase of training. This can be due to the more complex landscape of the loss function on the CIFAR10 dataset, so that it takes SGD several epochs to enter a basin of attraction of the global minimum. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our empirical findings strongly support the validity of the epochwise star-convex path of SGD in Definition 1. Therefore, Theorem 1 establishes an empirically-verified theory for characterizing the convergence property of SGD in training neural networks at an epoch level. In particular, it is well justified to successfully explain the stable epochwise convergence behavior observed in Figure 1. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/504d72f48c6e375be614122b28d35ba6f006dbaf2772a5ec1de37cc2abbedbc6.jpg",
+        "image_caption": [
+            "Figure 2: Verification of epochwise star-convex path. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
+            101,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 CONVERGENCE TO A GLOBAL MINIMIZER ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            388,
+            558,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The result developed in Theorem 1 shows that the variable sequence generated by SGD monotonically approaches a global minimizer at an epoch level. However, it does not guarantee the convergence of the variable sequence to a global minimizer (which requires the distance between SGD iterates and the global minimizer reduces to zero). We further explore such a convergence issue in the following two subsections. We first define a notion of an iterationwise star-convex path for SGD, based on which we formally establish the convergence of SGD to a global minimizer. Then, we provide empirical evidences to support the satisfaction of the iterationwise star-convex path by SGD. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 ITERATIONWISE STAR-CONVEX PATH ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We introduce the following definition of an iterationwise star-convex path for SGD. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Definition 2 (Iterationwise star-convex path). We call a path generated by SGD iterationwise starconvex if it satisfies: For all $k = 0 , 1 , \\ldots$ and for every ${ \\boldsymbol { x } } ^ { * } \\in \\mathcal { X } _ { \\xi _ { k } } ^ { * }$ , ",
+        "bbox": [
+            171,
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/269277c9638b6080b35e5eabe78a2d82295c4f7be16151c2b96f3c6130d729c3.jpg",
+        "text": "$$\n\\ell _ { \\xi _ { k } } ( x _ { k } ) - \\ell _ { \\xi _ { k } } ( x ^ { * } ) + \\langle x ^ { * } - x _ { k } , \\nabla \\ell _ { \\xi _ { k } } ( x _ { k } ) \\rangle \\leq 0 .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Compared to Definition 1 which defines the star-convex path of SGD at an epoch level, Definition 2 characterizes the star-convexity of SGD along the optimization path at a more refined iteration level. As we show in the result below, such a stronger property helps to regularize the convergence property of SGD at an iteration level, and is sufficient to guarantee convergence. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Theorem 3 (Convergence to global minimizer). Let Assumption 1 hold and apply SGD with learning rate $\\begin{array} { r } { \\eta < { \\frac { 1 } { L } } } \\end{array}$ to solve problem $( P )$ . Assume SGD follows an iterationwise star-convex path. Then, the sequence $\\{ x _ { k } \\} _ { k }$ generated by SGD converges to a global minimizer. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Theorem 3 formally establishes the convergence of SGD to a global minimizer along an iterationwise star-convex path. The main idea of the proof is to establish a consensus of the minimizing subsequences that are studied in Theorem 2, i.e., all these subsequences converge to the same limit – a common global minimizer of the loss functions over all the data samples. More specifically, our proof strategy consists of three steps: 1) show that every limit point of each subsequence is a common global minimizer; 2) prove that each subsequence has a unique limit point; 3) show that all these subsequences share the same unique limit point, which is a common global minimizer. We believe that the proof technique here can be of independent interest to the community. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Our analysis in Theorem 3 characterizes the intrinsic deterministic convergence property of SGD, which is an alternative view of the SGD path: It performs gradient descent on an individual loss component at each iteration. The star-convexity along the iteration path pushes the algorithm towards the common global minimizer. Such progress is shared across all data samples in every iteration and eventually leads to the convergence of SGD. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "We also note that the convergence result in Theorem 3 is based on a constant learning rate, which is typically used in practical training. This is very different from and much more desirable than the diminishing learning rate adopted in traditional analysis of SGD Nemirovski et al. (2009), which is a necessity to mitigate the negative effects caused by the variance of SGD. Furthermore, Theorem 3 shows that SGD converges to a common global minimizer where the gradient of loss function on all data samples vanish, and we therefore obtain the following interesting corollary. ",
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+    {
+        "type": "text",
+        "text": "Corollary 1 (Vanishing variance). Under the same settings of those of Theorem 3, the variance of stochastic gradients sampled by SGD converges to zero as iteration k goes to infinity. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Thus, upon convergence, the common global minimizer structure in deep learning leads to a selfvariance-reducing effect on SGD. Such a desirable effect is the core property of stochastic variancereduced algorithms that reduces sample complexity Johnson & Zhang (2013). Hence, this justifies in part that SGD is a sample-efficient algorithm in learning deep models. ",
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+    {
+        "type": "text",
+        "text": "DISCUSSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We want to mention that many nonconvex sensing models have an underlying true signal and hence naturally have common global minimizers, e.g., phase retrieval Zhang et al. (2017b), low-rank matrix recovery Tu et al. (2016), blind deconvolution Li et al. (2018), etc. This is also the case for some neural network sensing problems Zhong et al. (2017). Also, these problems have been shown to satisfy the so-called gradient dominance condition and the regularity condition locally around the global minimizers Zhou et al. (2016); Tu et al. (2016); Li et al. (2018); Zhong et al. (2017); Zhou & Liang (2017). These two geometric properties imply the star-convexity of the objective function, which necessarily imply the epochwise and iterationwise star-convex path of SGD. Therefore, our results also have implications on the convergence guarantee of SGD for solving these problems as well. ",
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+        "img_path": "images/a9aa078b572784a7e6a40bb7aee0272a907ca6a07fd29593831b62eadcfa463b.jpg",
+        "image_caption": [
+            "Figure 3: Verification of iterationwise star-convex path under crossentropy loss. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "4.2 VERIFYING ITERATIONWISE STAR-CONVEX PATH OF SGD ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this subsection, we conduct experiments to validate the iterationwise star-convex path of SGD introduced in Definition 2. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We train the aforementioned three types of neural networks, i.e., MLP, Alexnet and Inception, on CIFAR10 and MNIST datasets using SGD. The hyperparameter settings are the same as those mentioned in Section 3.1. We train these networks for a sufficient number of epochs to achieve a near-zero training loss. Due to the demanding requirement for storage, we record the variable sequence generated by SGD for all iterations in every tenth epoch, and set $x ^ { * }$ to be the final output of SGD. Then, for all the iterations in every tenth epoch, we evaluate the corresponding values of the terms on the left hand side in eq. (2) (denoted as $e _ { k }$ ). Then, we report the fraction of number of iterations that satisfy the iterationwise star-convexity (i.e., $e _ { k } < 0$ ) within such an epoch. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In all subfigures of Figure 3, the red curves denote the training loss and the blue bars denote the fraction of iterations that satisfy the iterationwise star-convexity within such an epoch. It can be seen from Figure 3 that, for all three networks on the MNIST dataset (second row), the path of SGD satisfies iterationwise star-convexity for most of the iterations, except in the last several epochs where the training loss (see the red curve) already well saturates at zero value. In fact, the convergence is typically observed well before such a point. This is because when the training loss is very close to the global minimum (i.e., the gradient is very close to zero), small perturbation of the landscape easily deviates the SGD path from the desired star-convexity. Hence, our experiments demonstrate that SGD follows the iterationwise star-convex path up to the convergence occurs. Furthermore, on the CIFAR10 dataset (first row of Figure 3), we observe a strong evidence for the iterationwise star-convex path of SGD after several epochs of the initial phase of training. This implies that the loss landscape on a more challenging dataset can be more complex. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Our empirical findings support the validity of the iterationwise star-convex path of SGD in a major part of practical training processes. Therefore, our convergence guarantee developed in Theorem 3 for SGD well justifies its practical success. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/b9d30d0acc1162b6b9829896132d8278b9aff55087890619dc53da069cd3d924.jpg",
+        "image_caption": [
+            "Figure 4: Iterationwise path on local minimum. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            395,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We next conduct further experiments to demonstrate that SGD follows the iterationwise star-convex path likely only for successful trainings to zero loss value, where a shared global minimum among all individual loss functions is achieved. To verify such a thought, we train an MLP using SGD on the CIFAR10 dataset under various settings with the number of hidden neurons ranging from 16 to 256. The results are shown in Figure 4, from which we observe that the training loss (i.e., red curves) converges to a non-zero value when the number of hidden neurons is small, implying that the algorithm likely attains a sub-optimal point which is not a common global minimum shared by all individual loss functions. In such trainings, we observe that the corresponding SGD paths have much fewer iterations satisfying the iterationwise star-convexity compared to the successful training instances shown in Figure 3. Thus, such empirical findings partially suggest that iterationwise star-convex SGD path more likely occurs when SGD can find a common global minimum, e.g., training overparameterized networks to zero loss value. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            863,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose an epochwise star-convex property of the optimization path of SGD, which we validate in various experiments. Based on such a property, we show that SGD approaches a global minimum at an epoch level. Then, we further examine the property at an iteration level, and empirically show that it is satisfied in a major part of training processes. As we prove theoretically, such a more refined property guarantees the convergence of SGD to a global minimum, and the algorithm enjoys a self-variance-reducing effect. We believe that our study sheds light on the success of SGD in training neural networks from both empirical aspect and theoretical aspect. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            194,
+            285,
+            209
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "A. Blum and R. L. Rivest. Training a 3-node neural network is NP-complete. In Proc. 1st Annual Workshop on Computational Learning Theory (COLT), pp. 9–18, 1988.   \nL. Bottou, F. E. Curtis, and J. Nocedal. Optimization methods for large-scale machine learning. ArXiv: 1606.04838, June 2016.   \nH. Daneshmand, J. M. Kohler, A. Lucchi, and T. Hofmann. Escaping saddles with stochastic gradients. In Proc. International Conference on Machine Learning (ICML), 2018.   \nA. Defazio, F. Bach, and S. Lacoste-Julien. SAGA: A fast incremental gradient method with support for non-strongly convex composite objectives. In Proc. Advances in Neural Information Processing Systems (NIPS), pp. 1646–1654. 2014.   \nJ. Duchi, E. Hazan, and Y. Singer. Adaptive subgradient methods for online learning and stochastic optimization. Journal of Machine Learning Research (JMLR), 12:2121–2159, July 2011.   \nR. Ge, F. Huang, C. Jin, and Y. Yuan. Escaping from saddle points — online stochastic gradient for tensor decomposition. In Proc. 28th Conference on Learning Theory (COLT), volume 40, pp. 797–842, 03–06 Jul 2015.   \nS. Ghadimi and G. Lan. Accelerated gradient methods for nonconvex nonlinear and stochastic programming. Mathematical Programming, 156(1):59–99, Mar 2016.   \nS. Ghadimi, G. Lan, and H. Zhang. Mini-batch stochastic approximation methods for nonconvex stochastic composite optimization. Mathematical Programming, 155(1):267–305, Jan 2016.   \nC. Jin, R. Ge, P. Netrapalli, S. M. Kakade, and M. I. Jordan. How to escape saddle points efficiently. In Proc. 34th International Conference on Machine Learning (ICML), volume 70, pp. 1724–1732, Aug 2017.   \nR. Johnson and T. Zhang. Accelerating stochastic gradient descent using predictive variance reduction. In Proc. 26th International Conference on Neural Information Processing Systems (NIPS), pp. 315–323, 2013.   \nD. Kingma and J. Ba. 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In Proc. 34th International Conference on Machine Learning (ICML), volume 70, pp. 4140–4149, Aug 2017.   \nY. Zhou and Y. Liang. Characterization of gradient dominance and regularity conditions for neural networks. ArXiv:1710.06910v2, Oct 2017.   \nY. Zhou and Y. Liang. Critical points of linear neural networks: Analytical forms and landscape properties. In Proc. International Conference on Learning Representations (ICLR), 2018.   \nY. Zhou, H. Zhang, and Y. Liang. Geometrical properties and accelerated gradient solvers of nonconvex phase retrieval. In Proc. 54th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 331–335, 2016.   \nY. Zhou, Z. Wang, and Y. Liang. Convergence of cubic regularization for nonconvex optimization under KL property. In Proc. Advances in Neural Information Processing Systems (NIPS), 2018. ",
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+    {
+        "type": "text",
+        "text": "Supplementary Materials ",
+        "text_level": 1,
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A PROOF OF THEOREM 1 ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Observe that the SGD update can be rewritten as the following optimization step ",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3a587c19061ea1b5e09d66df9118bd7f5f71c9dd3256b11c6774ae5c6712e04c.jpg",
+        "text": "$$\nx _ { k + 1 } = \\underset { u \\in \\mathbb { R } ^ { d } } { \\mathrm { a r g m i n } } \\Big \\{ \\underset { \\ b { \\xi } _ { k } } { \\ell } ( x _ { k } ) + \\langle u - x _ { k } , \\nabla \\ell _ { \\xi _ { k } } ( x _ { k } ) \\rangle + \\frac { 1 } { 2 \\eta } \\| u - x _ { k } \\| ^ { 2 } \\Big \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Note that the function $f _ { \\xi _ { k } } ( u )$ is linear, and we further obtain that for all $x ^ { \\ast } \\in \\mathcal { X } _ { \\xi _ { k } } ^ { \\ast }$ ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/fd05887e89e278aaac2900c9a3fee4c80ccdb0190ba47cf9517e92496404b5ba.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\eta \\big ( f _ { \\xi _ { k } } ( x _ { k + 1 } ) - f _ { \\xi _ { k } } ( x ^ { * } ) \\big ) = \\eta \\langle \\nabla \\ell _ { \\xi _ { k } } ( x _ { k } ) , x _ { k + 1 } - x ^ { * } \\rangle } & { } \\\\ { \\overset { ( i ) } { = } \\langle x _ { k } - x _ { k + 1 } , x _ { k + 1 } - x ^ { * } \\rangle } & { } \\\\ { } & { = \\cfrac { 1 } { 2 } \\Big ( \\| x _ { k } - x ^ { * } \\| ^ { 2 } - \\| x _ { k + 1 } - x ^ { * } \\| ^ { 2 } - \\| x _ { k + 1 } - x _ { k } \\| ^ { 2 } \\Big ) , } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where (i) uses the update rule of SGD. Rearranging the above inequality yields that ",
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+        "img_path": "images/2e3606006d867a9bf4605588f08057639a5dd968f063e9bcc8967045f02cd10b.jpg",
+        "text": "$$\nf _ { \\xi _ { k } } ( x _ { k + 1 } ) \\leq f _ { \\xi _ { k } } ( x ^ { * } ) + \\frac { 1 } { 2 \\eta } \\Big ( \\| x _ { k } - x ^ { * } \\| ^ { 2 } - \\| x _ { k + 1 } - x ^ { * } \\| ^ { 2 } - \\| x _ { k + 1 } - x _ { k } \\| ^ { 2 } \\Big ) .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "On the other hand, by smoothness of the loss function, we obtain that ",
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+    {
+        "type": "equation",
+        "img_path": "images/1b3bf5b20e4de2979aa019cdf935978c0533f225f304ca10f372602b663dd347.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\ell _ { \\xi _ { k } } ( x _ { k + 1 } ) \\leq \\ell _ { \\xi _ { k } } ( x _ { k } ) + \\langle x _ { k + 1 } - x _ { k } , \\nabla \\ell _ { \\xi _ { k } } ( x _ { k } ) \\rangle + \\displaystyle \\frac { L } { 2 } \\| x _ { k + 1 } - x _ { k } \\| ^ { 2 } } \\\\ & { \\qquad = f _ { \\xi _ { k } } ( x _ { k + 1 } ) + \\displaystyle \\frac { L } { 2 } \\| x _ { k + 1 } - x _ { k } \\| ^ { 2 } } \\\\ & { \\qquad \\overset { ( i ) } { \\leq } f _ { \\xi _ { k } } ( x ^ { * } ) + \\displaystyle \\frac { 1 } { 2 \\eta } ( \\| x _ { k } - x ^ { * } \\| ^ { 2 } - \\| x _ { k + 1 } - x ^ { * } \\| ^ { 2 } ) - ( \\displaystyle \\frac { 1 } { 2 \\eta } - \\displaystyle \\frac { L } { 2 } ) \\| x _ { k + 1 } - x _ { k } \\| ^ { 2 } } \\\\ & { \\qquad \\overset { ( i i ) } { \\leq } f _ { \\xi _ { k } } ( x ^ { * } ) + \\displaystyle \\frac { 1 } { 2 \\eta } ( \\| x _ { k } - x ^ { * } \\| ^ { 2 } - \\| x _ { k + 1 } - x ^ { * } \\| ^ { 2 } ) , } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where (i) follows from eq. (3) and (ii) is due to the choice of learning rate. Summing the above inequality over $k$ from $n B$ to $n ( B + 1 ) - 1$ yields that, for every $x ^ { * } \\in \\mathcal { X } ^ { * }$ in Definition 1, ",
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+        "img_path": "images/b5a925d5852d3d2eaa4c4be586e6c43ff6974c292046cfb62a33260b4cb891e3.jpg",
+        "text": "$$\n\\begin{array} { r } { \\| x _ { n ( B + 1 ) } - x ^ { * } \\| ^ { 2 } \\leq \\| x _ { n B } - x ^ { * } \\| ^ { 2 } - \\displaystyle \\sum _ { k = n B \\atop n ( B + 1 ) - 1 } ^ { n ( B + 1 ) - 1 } 2 \\eta \\big ( \\ell _ { \\xi _ { k } } ( x _ { k + 1 } ) - f _ { \\xi _ { k } } ( x ^ { * } ) \\big ) } \\\\ { \\leq \\| x _ { n B } - x ^ { * } \\| ^ { 2 } - \\displaystyle \\sum _ { k = n B } ^ { n ( B + 1 ) - 1 } 2 \\eta \\big ( \\ell _ { \\xi _ { k } } ( x _ { k + 1 } ) - \\ell _ { \\xi _ { k } } ( x ^ { * } ) \\big ) , } \\end{array}\n$$",
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+    {
+        "type": "text",
+        "text": "where the last inequality follows from the star-convex path of SGD in Definition 1. The desired result follows from the above inequality and the fact that $\\ell _ { \\xi _ { k } } ( x ^ { * } ) = \\operatorname* { i n f } _ { \\boldsymbol { u } \\in \\mathbb { R } ^ { d } } \\ell _ { \\xi _ { k } } ( \\boldsymbol { u } )$ for all $x ^ { * } \\in \\mathcal { X } ^ { * }$ Moreover, we conclude that the sequence $\\{ x _ { n B } \\} _ { B }$ is bounded. By continuity of $\\nabla \\ell _ { i }$ for all $i =$ $1 , . . . , n$ and the update rule of SGD, we further conclude that the entire sequence $\\{ \\boldsymbol { x } _ { k } \\} _ { k }$ is bounded. ",
+        "bbox": [
+            176,
+            681,
+            823,
+            738
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B PROOF OF THEOREM 2 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            757,
+            398,
+            773
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We first collect some facts. Recall that ${ { \\mathcal X } ^ { * } } = \\cap _ { i = 1 } ^ { n } { { \\mathcal X } _ { i } ^ { * } }$ is non-empty and bounded. Consider any fixed $t \\in \\{ 0 , \\ldots , n - 1 \\}$ and recall that $k = n B + t$ , $\\xi _ { k } = \\pi _ { B } ( t + 1 )$ . Then, one can check that the iterations $k$ with $\\xi _ { k } = \\dot { v } \\in \\{ 1 , 2 , . . . , n \\}$ form the subsequence $\\{ x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) - 1 } \\} _ { B }$ . ",
+        "bbox": [
+            173,
+            787,
+            825,
+            835
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Next, we prove item 1. Fix any $t \\in \\{ 0 , \\ldots , n - 1 \\}$ and sum eq. (4) over $k$ from $n B + t$ to $n ( B + 1 ) + t - 1$ yields that, for every $x ^ { * } \\in \\mathcal { X } ^ { * }$ in Definition 1, ",
+        "bbox": [
+            171,
+            840,
+            825,
+            871
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ff8fd3034ad715ffc2ad53d8642656e2347a5e6d780e5db40481dd37ca62950a.jpg",
+        "text": "$$\n\\| x _ { n ( B + 1 ) + t } - x ^ { * } \\| ^ { 2 } \\leq \\| x _ { n B + t } - x ^ { * } \\| ^ { 2 } - \\sum _ { k = n B + t } ^ { n ( B + 1 ) + t - 1 } 2 \\eta \\big ( \\ell _ { \\xi _ { k } } ( x _ { k + 1 } ) - f _ { \\xi _ { k } } ( x ^ { * } ) \\big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            232,
+            876,
+            766,
+            921
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Further summing the above inequality over $B$ from 0 to $K$ and rearranging, we obtain that ",
+        "bbox": [
+            171,
+            102,
+            767,
+            119
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/97e50eeb3d8a23c1d701ab1c218ab8e01e36f0c836a63597ed75379b0c30217a.jpg",
+        "text": "$$\n\\begin{array} { r l } { | x _ { n } ( K + 1 ) + t - x ^ { * } | | ^ { 2 } \\le \\| x _ { t } - x ^ { * } \\| ^ { 2 } - } & { \\displaystyle \\sum _ { k = n } ^ { n ( K + 1 ) - 1 } 2 \\eta \\big ( \\ell _ { \\xi _ { k } } ( x _ { k + 1 } ) - f _ { \\xi _ { k } } ( x ^ { * } ) \\big ) } \\\\ & { \\qquad - \\displaystyle \\sum _ { k = t } ^ { n - 1 } 2 \\eta \\big ( \\ell _ { \\xi _ { k } } ( x _ { k + 1 } ) - f _ { \\xi _ { k } } ( x ^ { * } ) \\big ) - \\displaystyle \\sum _ { k = n ( K + 1 ) } ^ { n ( K + 1 ) + t - 1 } 2 \\eta \\big ( \\ell _ { \\xi _ { k } } ( x _ { k + 1 } ) - f _ { \\xi _ { k } } ( x ^ { * } ) \\big ) , } \\\\ { \\le \\| x _ { t } - x ^ { * } \\| ^ { 2 } - } & { \\displaystyle \\sum _ { k = n } ^ { n ( K + 1 ) - 1 } 2 \\eta \\big ( \\ell _ { \\xi _ { k } } ( x _ { k + 1 } ) - \\ell _ { \\xi _ { k } } ( x ^ { * } ) \\big ) } \\\\ & { \\qquad - \\displaystyle \\sum _ { k = t } ^ { n - 1 } 2 \\eta \\big ( \\ell _ { \\xi _ { k } } ( x _ { k + 1 } ) - f _ { \\xi _ { k } } ( x ^ { * } ) \\big ) - \\displaystyle \\sum _ { k = n ( K + 1 ) } ^ { n ( K + 1 ) + t - 1 } 2 \\eta \\big ( \\ell _ { \\xi _ { k } } ( x _ { k + 1 } ) - f _ { \\xi _ { k } } ( x ^ { * } ) \\big ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            122,
+            821,
+            310
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where the last inequality follows from the the star-convex path of SGD in Definition 1. Consider the term $\\ell _ { \\xi _ { k } } ( x _ { k + 1 } ) - \\ell _ { \\xi _ { k } } ( x ^ { * } )$ in eq. (6) along the iterations with $\\xi _ { k } = v \\in \\{ 1 , 2 , . . . , n \\}$ . Such term can be rewritten as $\\ell _ { v } ( \\bar { x } _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } ) - \\ell _ { v } ( x ^ { * } )$ . Suppose for certain $v \\in \\{ 1 , 2 , . . . , n \\}$ the sequence $\\{ \\ell _ { v } ( x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } ) \\} _ { B }$ does not converge to its global minimum $\\operatorname* { i n f } _ { \\substack { x \\in \\mathbb { R } ^ { d } } } \\ell _ { v } ( x )$ . Then, by the cyclic sampling scheme with reshuffle, we conclude that the first summation term in eq. (6) diverges to $+ \\infty$ as $K  \\infty$ . Also, note that the last two summation terms have finite number of elements, which are all bounded as $\\{ \\boldsymbol { x } _ { k } \\} _ { k }$ is bounded. Therefore, we conclude that the sequences $\\{ x _ { n B + t } \\} _ { B }$ for $t = 0 , . . . , n - 1$ converge to $x ^ { * }$ for all candidates $x ^ { * } \\in \\mathcal { X } ^ { * }$ in Definition 1. Next, consider the case in which there are multiple such candidate $x ^ { * } s$ in Definition 1. Then, the previous sentence states that $\\{ x _ { n B + t } \\} _ { B }$ converges to multiple limits, which cannot happen for a convergent sequence. This leads to a contradiction. Consider the other case that there is only one such candidate $x ^ { * }$ in Definition 1. Then, we conclude that all the sequences $\\{ x _ { n B + t } \\} _ { B }$ for $t = 0 , . . . , n - 1$ converge to $x ^ { * }$ , i.e., the entire sequence $\\{ x _ { k } \\} _ { k }$ converges to such unique common global minimizer. This contradicts with our assumption that $\\{ \\ell _ { v } ( x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } ) \\} _ { B }$ does not converge to $\\operatorname* { i n f } _ { \\substack { x \\in \\mathbb { R } ^ { d } } } \\ell _ { v } ( x )$ for certain $v$ . Combining both cases, we obtain the desired claim of item 1. ",
+        "bbox": [
+            173,
+            321,
+            825,
+            542
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Next, we prove item 2. Note that sequence $\\{ x _ { k } \\} _ { k }$ is bounded . Fix any $v \\in \\{ 1 , \\ldots , n \\}$ and consider any limit point $z _ { v }$ of $\\{ x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } \\} _ { B }$ , i.e., $x _ { n B _ { j } + \\pi _ { B _ { j } } ^ { - 1 } ( v ) } \\stackrel { j } { \\to } z _ { v }$ along a proper subsequence. From item 1 we know that $\\begin{array} { r } { \\ell _ { v } \\big ( x _ { n B _ { j } + \\pi _ { B _ { j } } ^ { - 1 } ( v ) } \\big ) \\stackrel { j } { \\to } \\operatorname* { i n f } _ { x \\in \\mathbb { R } ^ { d } } \\ell _ { v } ( x ) } \\end{array}$ . This, together with the continuity of the loss function, implies that $z _ { v } \\in \\mathcal { X } _ { v } ^ { * }$ for all $v$ . ",
+        "bbox": [
+            173,
+            547,
+            826,
+            630
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C PROOF OF THEOREM 3 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            647,
+            398,
+            664
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Recall that eq. (4) shows that, for all $x ^ { \\ast } \\in \\mathcal { X } _ { \\xi _ { k } } ^ { \\ast }$ ",
+        "bbox": [
+            174,
+            678,
+            485,
+            694
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/90c75b1773f750353ea0d8a3bc7197f8947b60358914fe1d1c121116ffbd509a.jpg",
+        "text": "$$\n\\begin{array} { r l r } & { } & { \\ell _ { \\xi _ { k } } ( x _ { k + 1 } ) \\le f _ { \\xi _ { k } } ( x ^ { * } ) + \\displaystyle \\frac { 1 } { 2 \\eta } \\Big ( \\| x _ { k } - x ^ { * } \\| ^ { 2 } - \\| x _ { k + 1 } - x ^ { * } \\| ^ { 2 } \\Big ) } \\\\ & { } & { \\stackrel { ( i ) } { \\le } \\ell _ { \\xi _ { k } } ( x ^ { * } ) + \\displaystyle \\frac { 1 } { 2 \\eta } \\Big ( \\| x _ { k } - x ^ { * } \\| ^ { 2 } - \\| x _ { k + 1 } - x ^ { * } \\| ^ { 2 } \\Big ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            297,
+            696,
+            697,
+            765
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where (i) follows from the iterationwise star-convex path in Definition 2. Since $\\ell _ { \\xi _ { k } } ( x _ { k + 1 } ) - \\ell _ { \\xi _ { k } } ( x ^ { * } ) \\geq$ 0, we conclude that for all $k = 0 , 1 , \\ldots$ and every $x ^ { \\ast } \\in \\mathcal { X } _ { \\xi _ { k } } ^ { \\ast }$ , ",
+        "bbox": [
+            173,
+            765,
+            823,
+            795
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7be869b5f09f40cadf53ad6fc80f0d093da15b5f2618830e5db79bb9e8969b14.jpg",
+        "text": "$$\n\\lVert x _ { k + 1 } - x ^ { * } \\rVert \\leq \\lVert x _ { k } - x ^ { * } \\rVert .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            405,
+            797,
+            591,
+            815
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Next, consider any $v \\in \\{ 1 , . . . , n \\}$ , we show that every limit point $z _ { v }$ of $\\{ x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } \\} _ { B }$ is in $\\mathcal { X } ^ { \\ast }$ . By eq. (9), we know that $\\{ \\| x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } - x ^ { * } \\| \\} _ { B }$ is decreasing. Consider a limit point $z _ { v }$ associated with the subsequence such that $x _ { n B _ { j } + \\pi _ { B _ { j } } ^ { - 1 } ( v ) } \\to z _ { v }$ . Then, for all $B _ { j } \\geq B$ we know that, for any fixed $x ^ { * } \\in \\mathcal { X } ^ { * }$ , ",
+        "bbox": [
+            173,
+            823,
+            826,
+            892
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5d6f72429a7aa17ccefc3523eef60ddb1630089d379d8bf924d6f5225ecbf79a.jpg",
+        "text": "$$\n\\| z _ { v } - x ^ { * } \\| \\overset { j } {  } \\| x _ { n B _ { j } + \\pi _ { B _ { j } } ^ { - 1 } ( v ) } - x ^ { * } \\| \\leq \\| x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } - x ^ { * } \\| .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            302,
+            893,
+            696,
+            922
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Next, we prove by contradiction. Suppose that $z _ { v } \\notin \\mathcal { X } ^ { \\ast }$ . Then, eq. (10) implies that $\\parallel x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } -$ $x ^ { * } \\Vert > 0$ for all large $B$ and any $x ^ { \\ast } \\in \\mathcal { X } ^ { \\ast }$ . Combining this conclusion with item 2 of Theorem 2, it follows that all the limit points of $\\{ x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } \\} _ { B }$ are in ${ \\mathcal { X } } _ { v } ^ { * } \\mid { \\mathcal { X } } ^ { * }$ . Since ${ { \\mathcal X } ^ { * } } = \\cap _ { i = 1 } ^ { n } { { \\mathcal X } _ { i } ^ { * } }$ , it follows that the limit points of $\\{ x _ { n B + \\pi _ { B } ^ { - 1 } ( u ) } \\} _ { B }$ are different from those of $\\{ x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } \\} _ { B }$ for any $u \\ne v$ . Now consider a subsequence $x _ { n B _ { j } + \\pi _ { B _ { j } } ^ { - 1 } ( v ) } \\to z _ { v } \\in \\mathcal { X } _ { v } ^ { * } \\setminus \\mathcal { X } ^ { * }$ . Also, consider the subsequence $\\{ x _ { n B _ { j } + \\pi _ { B _ { j } } ^ { - 1 } ( u ) } \\} _ { j }$ with bseq $u \\ne v$ he  of lic samplisuch that hat with probability(this occurs with a $B _ { j ( s ) }$ $B _ { j }$ $\\pi _ { B _ { j ( s ) } } ^ { - 1 } ( u ) = \\pi _ { B _ { j ( s ) } } ^ { - 1 } ( v ) + 1$ constant probability in every epoch). Applying eq. (9) along this subsequence, we conclude that ",
+        "bbox": [
+            173,
+            102,
+            826,
+            247
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/30f329b19c12d413967d2620834a326d8f30094a6548a709e57715346fad2f99.jpg",
+        "text": "$$\n\\begin{array} { r } { \\| x _ { n B _ { j ( s ) } + \\pi _ { B _ { j ( s ) } } ^ { - 1 } ( u ) } - z _ { v } \\| \\leq \\| x _ { n B _ { j ( s ) } + \\pi _ { B _ { j ( s ) } } ^ { - 1 } ( v ) } - z _ { v } \\| \\overset { j } {  } 0 . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            302,
+            263,
+            694,
+            294
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Let j → ∞ in the above equation, we conclude that xnBj(s)+π−1Bj(s)(u ) → zv , i.e., zv is a limit point of $\\{ x _ { n B _ { j } + \\pi _ { B _ { j } } ^ { - 1 } ( u ) } \\} _ { B }$ . Note that $z _ { v }$ is a limit point of $\\{ x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } \\} _ { B }$ . This contradicts our previous conclusion that the limit points of $\\{ x _ { n B + \\pi _ { B } ^ { - 1 } ( u ) } \\} _ { B }$ must be different from those of $\\{ x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } \\} _ { B }$ for $u \\ne v$ . Thus, we must have for all $v = 1 , \\ldots , n$ , every limit point $z _ { v }$ of $\\{ x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } \\} _ { B }$ is in $\\mathcal { X } ^ { \\ast }$ Then, by eq. (9) we further conclude that for all $B = 0 , 1 , . . .$ . ",
+        "bbox": [
+            173,
+            309,
+            826,
+            404
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bafd5dda2306973512203592d51840cd30989e85dc7f1980aec24a6a5b51661f.jpg",
+        "text": "$$\n\\| x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } - z _ { v } \\| \\leq \\| x _ { n ( B - 1 ) + \\pi _ { B - 1 } ^ { - 1 } ( v ) } - z _ { v } \\| .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            334,
+            420,
+            661,
+            443
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Note that $x _ { n B _ { j } + \\pi _ { B _ { j } } ^ { - 1 } ( v ) } \\overset { j } { \\to } z _ { v }$ . Thus, for all $B \\geq B _ { j }$ the above inequality implies that ",
+        "bbox": [
+            173,
+            462,
+            732,
+            487
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4ea5977446fc858ee044f5b6499c380aded76798999709c7ac2a632294467824.jpg",
+        "text": "$$\n\\begin{array} { r } { \\| x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } - z _ { v } \\| \\le \\| x _ { n B _ { j } + \\pi _ { B _ { j } } ^ { - 1 } ( v ) } - z _ { v } \\| \\overset { j } {  } 0 . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            507,
+            665,
+            535
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "This shows that $\\{ x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } \\} _ { B }$ has a unique limit point $z _ { v }$ , which is an element of $\\mathcal { X } ^ { \\ast }$ . Next, consider the limits $z _ { u } , z _ { v } ( u \\ne v )$ of the sequences $\\{ x _ { n B + \\pi _ { B } ^ { - 1 } ( u ) } \\} _ { B } , \\{ x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } \\} _ { B }$ , respectively. By eq. (9) and the fact that $z _ { u } \\in \\mathcal { X } ^ { * }$ , we conclude that ",
+        "bbox": [
+            173,
+            553,
+            828,
+            603
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b34ea88ace14deec0010e73ccf6c489650ed2929d93c0e4f8888a83b4cc78236.jpg",
+        "text": "$$\n\\begin{array} { r } { \\| x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) } - z _ { u } \\| \\le \\| x _ { n ( B - 1 ) + \\pi _ { B - 1 } ^ { - 1 } ( v ) } - z _ { u } \\| \\le \\| x _ { n ( B - 2 ) + \\pi _ { B - 2 } ^ { - 1 } ( u ) } - z _ { u } \\| \\overset { B } { \\to } 0 . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            222,
+            622,
+            774,
+            647
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Thus, $x _ { n B + \\pi _ { B } ^ { - 1 } ( v ) }  z _ { u }$ , and we conclude that $z _ { v } = z _ { u }$ for all $v \\neq u$ , i.e., the whole sequence $\\{ x _ { k } \\}$ has a unique limit point in $\\mathcal { X } ^ { \\ast }$ . ",
+        "bbox": [
+            171,
+            665,
+            823,
+            698
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "D SUPPLEMENTARY EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            729,
+            482,
+            746
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In this section, we provide more experiments to illustrate the star-convexity property of the SGD path from other different aspects. ",
+        "bbox": [
+            173,
+            768,
+            823,
+            799
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Verification of epochwise star-convexity with different reference points ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            804,
+            666,
+            820
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In Figure 5 we verify the epochwise star-convexity of the SGD path by setting the reference point $x ^ { * }$ to be the output of SGD at different intermediate epochs (i.e., 60,80,100,120 epochs), where the SGD has already saturated close to zero loss. The experiments are conducted by training the Alexnet and MLP on Cifar10 using SGD. As can be seen from Figure 5, the epochwise star-convexity still hold (i.e., $e _ { B } < 0$ ) after certain epochs in the initial training phase. This shows that the observed star-convex path does not depend on the choice of reference point (so long as they achieve near-zero loss). ",
+        "bbox": [
+            173,
+            825,
+            825,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/47b6aa52170926d76249d7e2630f5047ce2998413c192414bcaf31d053cf811c.jpg",
+        "image_caption": [
+            "Figure 5: Verification of epochwise star-convexity when the reference point $x ^ { * }$ is taken at end of different epochs. Top plots correspond to the training of Alexnet with $x ^ { * }$ taken at 60th, 80th, 100th, and 120th epochs, and bottom plots correspond to the training of MLP with $x ^ { * }$ taken at 60th, 80th, $1 0 0 \\mathrm { { t h } }$ , and $1 2 0 \\mathrm { { t h } }$ epochs. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            99,
+            803,
+            299
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Growth of weight norm in neural network training ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            391,
+            529,
+            406
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In Figure 6, we present the growth of the $\\ell _ { 2 }$ norm of network weights in training different neural networks on Cifar10 under the cross-entropy loss. It can be seen from the figure that the norm of the weights increases slowly (logarithmly) after the training loss achieves near-zero. This is because the gradient is nearly zero when the training is close to the global minimum, and therefore the updates of the weights are very small. ",
+        "bbox": [
+            173,
+            412,
+            825,
+            482
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/1d3f0254a74c42f2b28e56e6beee460c762a4e53e11a64623288eb09a4b60294.jpg",
+        "image_caption": [
+            "Figure 6: $\\ell _ { 2 }$ norm of network weights and training loss of different networks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
+            496,
+            805,
+            618
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Verification of star-convexity under MSE loss ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            667,
+            488,
+            683
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In Figure 7, we verify the epochwise star-convexity by training different networks on MNIST dataset under the MSE loss (i.e., $\\ell _ { 2 }$ loss). We note that unlike the cross-entropy loss, zero value can be achieved by the MSE loss. We set the reference point $x ^ { * }$ to be the output of SGD at the 40th epoch. It can be seen from the figure that the residue $e _ { B }$ is negative along the entire optimization path, demonstrating that the SGD path satisfies the epochwise star-convexity under the MSE loss. ",
+        "bbox": [
+            174,
+            690,
+            826,
+            760
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/6aba459d7f1438c7676dbf6225b82fff306f10712555be2b4ebc437c352d2b13.jpg",
+        "image_caption": [
+            "Figure 7: Verification of epochwise star-convexity under MSE loss (i.e., $\\ell _ { 2 }$ loss). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
+            773,
+            794,
+            893
+        ],
+        "page_idx": 13
+    }
+]

parse/train/S1lN69AT-/S1lN69AT-.md

@@ -0,0 +1,146 @@
+# TO PRUNE, OR NOT TO PRUNE: EXPLORING THE EFFICACY OF PRUNING FOR MODEL COMPRESSION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Model pruning seeks to induce sparsity in a deep neural network’s various connection matrices, thereby reducing the number of nonzero-valued parameters in the model. Recent reports (Han et al., 2015a; Narang et al., 2017) prune deep networks at the cost of only a marginal loss in accuracy and achieve a sizable reduction in model size. This hints at the possibility that the baseline models in these experiments are perhaps severely over-parameterized at the outset and a viable alternative for model compression might be to simply reduce the number of hidden units while maintaining the model’s dense connection structure, exposing a similar trade-off in model size and accuracy. We investigate these two distinct paths for model compression within the context of energy-efficient inference in resource-constrained environments and propose a new gradual pruning technique that is simple and straightforward to apply across a variety of models/datasets with minimal tuning and can be seamlessly incorporated within the training process. We compare the accuracy of large, but pruned models (large-sparse) and their smaller, but dense (small-dense) counterparts with identical memory footprint. Across a broad range of neural network architectures (deep CNNs, stacked LSTM, and seq2seq LSTM models), we find large-sparse models to consistently outperform small-dense models and achieve up to $1 0 \mathrm { x }$ reduction in number of non-zero parameters with minimal loss in accuracy.
+
+# 1 INTRODUCTION
+
+Over the past few years, deep neural networks have achieved state-of-the-art performance on several challenging tasks in the domains of computer vision, speech recognition, and natural language processing. Driven by increasing amounts of data and computational power, deep learning models have become bigger and deeper to better learn from data. While these models are typically deployed in a datacenter back-end, preserving user privacy and reducing user-perceived query times mandate the migration of the intelligence offered by these deep neural networks towards edge computing devices. Deploying large, accurate deep learning models to resource-constrained computing environments such as mobile phones, smart cameras etc. for on-device inference poses a few key challenges. Firstly, state-of-the-art deep learning models routinely have millions of parameters requiring \~MBs of storage, whereas on-device memory is limited. Furthermore, it is not uncommon for even a single model inference to invoke \~billions of memory accesses and arithmetic operations, all of which consume power and dissipate heat which may drain the limited battery capacity and/or test the device’s thermal limits.
+
+Confronting these challenges, a growing body of work has emerged that intends to discover methods for compressing neural network models while limiting any potential loss in model quality. Latencysensitive workloads relying on energy-efficient on-device neural network inference are often memory bandwidth-bound, and model compression offers the two-fold benefit of reducing the total number of energy-intensive memory accesses as well as improving the inference time due to an effectively higher memory bandwidth for fetching compressed model parameters. Within the realm of model compression techniques, pruning away (forcing to zero) the less salient connections (parameters) in the neural network has been shown to reduce the number of nonzero parameters in the model with little to no loss in the final model quality. Model pruning enables trading off a small degradation in model quality for a reduction in model size, potentially reaping improvements in inference time and energy-efficiency. The resulting pruned model typically has sparse connection matrices, so efficient inference using these sparse models requires purpose-built hardware capable of loading sparse matrices and/or performing sparse matrix-vector operations (Zhang et al., 2016; Han et al., 2016; Parashar et al., 2017). Also, representing sparse matrices carries with it an additional storage overhead increasing the model’s net memory footprint which must also be taken into consideration.
+
+In this work, we perform a closer examination of the effectiveness of model pruning as a means for model compression. From the perspective of on-device neural network inference, given a bound on the model’s memory footprint, how can we arrive at the most accurate model? We aim to answer this question by comparing the quality of the models obtained through two distinct methods: (1) training a large model, but pruned to obtain a sparse model with a small number of nonzero parameters (large-sparse); and (2) training a small-dense model with size comparable to the large-sparse model. Both of these methods expose a model accuracy and size tradeoff, but differ remarkably in terms of their implications on the design of the underlying hardware architecture. For this comparative study, we pick models across a diverse set of application domains: InceptionV3 (Szegedy et al., 2016) and MobileNets (Howard et al., 2017) for image recognitions tasks, stacked LSTMs for language modeling, and seq2seq models used in Google’s Neural Machine Translation (Wu et al., 2016) system. In the process of this investigation, we also develop a simple gradual pruning approach that requires minimal tuning and can be seamlessly incorporated within the training process and demonstrate its applicability and performance on an assortment of neural network architectures.
+
+# 2 RELATED WORK
+
+Early works in the 1990s (LeCun et al., 1990; Hassibi et al., 1993) performed pruning using a second-order Taylor approximation of the increase in the loss function of the network when a weight is set to zero. In Optimal Brain Damage (LeCun et al., 1990), the saliency for each weight was computed using a diagonal Hessian approximation, and the low-saliency weights were pruned from the network and the network was retrained. In Optimal Brain Surgeon (Hassibi et al., 1993), the saliency for each weight was computed using the inverse Hessian matrix, and the low-saliency weights were pruned and all other weights in the network were updated using the Hessian matrix.
+
+More recently, magnitude-based weight pruning methods have become popular techniques for network pruning (Han et al., 2015b;a; See et al., 2016; Narang et al., 2017). Magnitude-based weight pruning techniques are computationally efficient, scaling to large networks and datasets. Our automated gradual pruning algorithm prunes the smallest magnitude weights to achieve a preset level of network sparsity. In contrast with the works listed above, our paper focuses on comparing the model accuracy and size tradeoff of large-sparse versus small-dense models.
+
+A work similar to ours is the work by Narang et al. (2017) on pruning a RNN and GRU model for speech recognition and showing that a sparse RNN that was pruned outperformed a dense RNN trained normally of comparable size. While they provide one data point comparing the performance of a sparse vs dense model, our work does an extensive comparison of sparse vs dense models across a wide range of models in different domains (vision and NLP). Narang et al. also introduce a gradual pruning scheme based on pruning all the weights in a layer less than some threshold (manually chosen) which is linear with some slope in phase 1 and linear with some slope in phase 2 followed by normal training. Compared to their approach, we do not have two phases and do not have to choose two slopes, and we do not need to choose weight thresholds for each layer (we rely on a sparsity schedule which determines the weight thresholds). Thus, our technique is simpler, doesn’t require much hyperparameter tuning, and is shown to perform well across different models.
+
+Within the context of reducing model size by removing redundant connections, several recent works (Anwar et al., 2015; Lebedev & Lempitsky, 2015; Li et al., 2016; Changpinyo et al., 2017) propose techniques to prune and induce sparsity in a structured way, motivated primarily by the desire to speedup computations on existing hardware architectures optimized for dense linear algebra. Such techniques perform coarse-grain pruning and depend critically on the structure of the convolutional layers, and may not be directly extensible to other neural network architectures that lack such structural properties (LSTMs for instance). On the contrary, our method does not make any assumptions about the structure of the network or its constituent layers and is therefore more generally applicable.
+
+While pruning focuses on reducing the number of non-zero parameters, in principle, model pruning can be used in conjunction with other techniques to further reduce model size. Quantization techniques aim to reduce the number of bits required to represent each parameter from 32-bit floats to 8 bits or fewer. Different quantization techniques such as fixed-point quantization (Vanhoucke et al., 2011) or vector quantization (Gong et al., 2014) achieve different compression ratios and accuracies but also require different software or hardware to support inference at runtime. Pruning can be combined with quantization to achieve maximal compression (Han et al., 2015a). In addition, an emerging area of research is low precision networks where the parameters and/or activations are quantized to 4 bits or fewer (Courbariaux et al., 2015; Lin et al., 2015; Hubara et al., 2016; Rastegari et al., 2016; Zhu et al., 2016). Besides quantization, other potentially complementary approaches to reducing model size include low-rank matrix factorization (Denil et al., 2013; Denton et al., 2014; Jaderberg et al., 2014; Lebedev et al., 2014) and group sparsity regularization to arrive at an optimal layer size (Alvarez & Salzmann, 2016).
+
+![](images/b8a464b65ac8989f1c3928533cfcfc3b5ca03050aa33b8deb596993adf0875b5.jpg)  
+Figure 1: Sparsity function used for gradual pruning
+
+Table 1: Model size and accuracy tradeoff for sparse-InceptionV3   
+
+<table><tr><td>Sparsity</td><td>NNZ params</td><td>Top-1 acc.</td><td>Top-5 acc.</td></tr><tr><td>0%</td><td>27.1M</td><td>78.1%</td><td>94.3%</td></tr><tr><td>50%</td><td>13.6M</td><td>78.0%</td><td>94.2%</td></tr><tr><td>75%</td><td>6.8M</td><td>76.1%</td><td>93.2%</td></tr><tr><td>87.5%</td><td>3.3M</td><td>74.6%</td><td>92.5%</td></tr></table>
+
+# 3 METHODS
+
+We extend the TensorFlow (Abadi et al., 2015) framework to prune the network’s connections during training. For every layer chosen to be pruned, we add a binary mask variable which is of the same size and shape as the layer’s weight tensor and determines which of the weights participate in the forward execution of the graph. We inject ops into the TensorFlow training graph to sort the weights in that layer by their absolute values and mask to zero the smallest magnitude weights until some desired sparsity level $s$ is reached. The back-propagated gradients flow through the binary masks, and the weights that were masked in the forward execution do not get updated in the backpropagation step. We introduce a new automated gradual pruning algorithm in which the sparsity is increased from an initial sparsity value $s _ { i }$ (usually 0) to a final sparsity value $s _ { f }$ over a span of $n$ pruning steps, starting at training step $t _ { 0 }$ and with pruning frequency $\Delta t$ :
+
+$$
+s _ { t } = s _ { f } + ( s _ { i } - s _ { f } ) \left( 1 - \frac { t - t _ { 0 } } { n \Delta t } \right) ^ { 3 } \mathrm { ~ f o r ~ } t \in \{ t _ { 0 } , ~ t _ { 0 } + \Delta t , ~ . . . , ~ t _ { 0 } + n \Delta t \}
+$$
+
+The binary weight masks are updated every $\Delta t$ steps as the network is trained to gradually increase the sparsity of the network while allowing the network training steps to recover from any pruninginduced loss in accuracy. In our experience, varying the pruning frequency $\Delta t$ between 100 and 1000 training steps had a negligible impact on the final model quality. Once the model achieves the target sparsity $s _ { f }$ , the weight masks are no longer updated. The intuition behind this sparsity function in equation (1) is to prune the network rapidly in the initial phase when the redundant connections are abundant and gradually reduce the number of weights being pruned each time as there are fewer and fewer weights remaining in the network, as illustrated in Figure 1. In the experimental results presented in this paper, pruning is initiated after the model has been trained for a few epochs or from a pre-trained model. This determines the value for the hyperparameter $t _ { 0 }$ . A suitable choice for $n$ is largely dependent on the learning rate schedule. Stochastic gradient descent (and its many variants) typically decay the learning rate during training, and we have observed that pruning in the presence of an exceedingly small learning rate makes it difficult for the subsequent training steps to recover from the loss in accuracy caused by forcing the weights to zero. At the same time, pruning with too high of a learning rate may mean pruning weights when the weights have not yet converged to a good solution, so the pruning schedule should be chosen closely with the learning rate schedule.
+
+![](images/7fbebfe6d2a1ec0deba98179b35da1400747e9503979eb08eb7c1464ff47b4e3.jpg)  
+Figure 2: (a) The gradual sparsity function and exponentially decaying learning rate used for training sparse-InceptionV3 models. (b) Evolution of the model’s accuracy during the training process
+
+![](images/8e5c10d27205284de0780dc0e30b8cb8150f92a11177d9cc85416f827808d715.jpg)  
+Figure 3: MobileNet sparse vs dense results
+
+![](images/702b771ad2805dc43f0e503ff094d90992d665a3ee98080cd91e746fe237d7a2.jpg)  
+Figure 4: PTB sparse vs dense results
+
+Figure 2a shows the learning rate and the pruning schedule used for training sparse-InceptionV3 (Szegedy et al., 2016) models. All the convolutional layers in this model are pruned using the same sparsity function, and pruning occurs in the regime where the learning rate is still reasonably high to allow the network to heal from the pruning-induced damage. Figure 2b offers more insight into how this pruning scheme interacts with the training procedure. For the $8 7 . 5 \%$ sparse model, with the gradual increase in sparsity, there comes a point when the model suffers a near-catastrophic degradation, but recovers nearly just as quickly with continued training. This behavior is more pronounced in the models trained to have higher sparsity. Table 1 compares the performance of sparse-InceptionV3 models pruned to varying extents. As expected, there is a gradual degradation in the model quality as the sparsity increases. However, a $50 \%$ sparse model performs just as well as the baseline $0 \%$ sparsity), and there is only a $2 \%$ decrease in top-5 classification accuracy for the $8 7 . 5 \%$ sparse model which offers an ${ 8 } \mathbf { { x } }$ reduction in number of nonzero (NNZ) model parameters. Also note that since the weights are initialized randomly, the sparsity in the weight tensors does not exhibit any specific structure. Furthermore, the pruning method described here does not depend on any specific property of the network or the constituent layers, and can be extended directly to a wide-range of neural network architectures.
+
+# 4 COMPARING large-sparse AND small-dense MODELS
+
+# 4.1 MOBILENETS
+
+MobileNets are a class of efficient convolutional neural networks designed specifically for mobile vision applications (Howard et al., 2017). Instead of using standard convolutions, MobileNets are based on a form of factorized convolutions called depthwise separable convolution. Depthwise separable convolutions consist of a depthwise convolution followed by a 1x1 convolution called a pointwise convolution. This factorization significantly reduces the number of parameters in the model by filtering and combining input channels in two separate steps instead of together as in the standard convolution. The MobileNet architecture consists of one standard convolution layer acting on the input image, a stack of depthwise separable convolutions, and finally averaging pooling and fully connected layers. For the dense baseline model with width multiplier 1.0, there are a total of 4.21M parameters, $9 9 \%$ of which are in the 1x1 pointwise convolution layers $( 7 4 . 6 \% )$ and fully connected layers $( 2 4 . 3 \% )$ . We do not prune the parameters in the one standard convolution layer and in the depthwise convolution layers since there are very few parameters in those layers $( 1 . 1 \% )$ .
+
+Table 2: MobileNet sparse vs dense results   
+
+<table><tr><td>Width</td><td>Sparsity</td><td>NNZ params</td><td>Top-1 acc.</td><td>Top-5 acc.</td></tr><tr><td>0.25</td><td>0%</td><td>0.46M</td><td>50.6%</td><td>75.0%</td></tr><tr><td>0.5</td><td>0%</td><td>1.32M</td><td>63.7%</td><td>85.4%</td></tr><tr><td>0.75</td><td>0%</td><td>2.57M</td><td>68.4%</td><td>88.2%</td></tr><tr><td rowspan="4">1.0</td><td>0%</td><td>4.21M</td><td>70.6%</td><td>89.5%</td></tr><tr><td>50%</td><td>2.13M</td><td>69.5%</td><td>89.5%</td></tr><tr><td>75%</td><td>1.09M</td><td>67.7%</td><td>88.5%</td></tr><tr><td>90%</td><td>0.46M</td><td>61.8%</td><td>84.7%</td></tr><tr><td></td><td>95%</td><td>0.25M</td><td>53.6%</td><td>78.9%</td></tr></table>
+
+Table 3: PTB sparse vs dense results   
+
+<table><tr><td>Model</td><td>Sparsity</td><td>NNZ params</td><td>Per- plexity</td></tr><tr><td>Small</td><td>0%</td><td>4.6M</td><td>115.30</td></tr><tr><td rowspan="5">Medium</td><td>0%</td><td>19.8M</td><td>83.37</td></tr><tr><td>80%</td><td>4.0M</td><td>83.87</td></tr><tr><td>85%</td><td>3.0M</td><td>85.17</td></tr><tr><td>90%</td><td>2.0M</td><td>87.86</td></tr><tr><td>95% 97.5%</td><td>1.0M 0.5M</td><td>96.30 113.6</td></tr><tr><td rowspan="5">Large</td><td>0%</td><td>66M</td><td>78.45</td></tr><tr><td>80%</td><td>13.2M</td><td>77.52</td></tr><tr><td>85%</td><td>9.9M</td><td>78.31</td></tr><tr><td>90%</td><td>6.6M</td><td>80.24</td></tr><tr><td>95%</td><td>3.3M</td><td>87.83</td></tr><tr><td></td><td>97.5%</td><td>1.7M</td><td>103.20</td></tr></table>
+
+The width multiplier is a parameter of the MobileNet network that allows trading off the accuracy of the model with the number of parameters and computational cost. The width multiplier of the baseline model is 1.0. For a given width multiplier $\alpha \in ( 0 , 1 ]$ , the number of input channels and the number of output channels in each layer is scaled by $\alpha$ relative to the baseline 1.0 model. We compare the performance of dense MobileNets trained with width multipliers 0.75, 0.5, and 0.25 with the performance of sparse MobileNets pruned from dense 1.0 MobileNet in Figure 3 and Table 2 on the ImageNet dataset. We see that for a given number of non-zero parameters, sparse MobileNets are able to outperform dense MobileNets. For example, the $7 5 \%$ sparse model (which has 1.09 million parameters and a top-1 accuracy of $6 7 . 7 \%$ ) outperforms the dense 0.5 MobileNet (which has 1.32 million parameters and a top-1 accuracy of $6 3 . 7 \%$ ) by $4 \%$ in top-1 accuracy while being smaller. Similarly, the $90 \%$ sparse model (which has 0.46 million parameters and a top-1 accuracy of $6 1 . 8 \%$ ) outperforms the dense 0.25 MobileNet (which has 0.46 million parameters and a top-1 accuracy of $5 0 . 6 \%$ by $1 0 . 2 \%$ in top-1 accuracy while having the same number of non-zero parameters.
+
+Overall, pruning is a promising approach for model compression even for an architecture that was designed to be compact and efficient by using depthwise separable convolutions instead of standard convolutions as a factorization-like technique to reduce the number of parameters. The sparsity parameter is shown to be an effective way to trade off the accuracy of a model with its memory usage and compares favorably with the width multiplier in MobileNet. Training a sparse MobileNet using our gradual pruning algorithm is also easy. For pruning a dense MobileNet, we used the same learning rate schedule as for training a dense MobileNet but with an initial learning rate 10 times smaller than for training a dense MobileNet, and all other hyperparameters were kept the same.
+
+# 4.2 PENN TREE BANK (PTB) LANGUAGE MODEL
+
+We train an LSTM language model on the Penn Tree Bank dataset using the models and training procedure described in Zaremba et al. (2014). At each time step, the LSTM language model outputs the probability of the next word in the sentence given the history of previous words. The loss function is the average negative log probability of the target words, and the perplexity is the exponential of the loss function. The language model is composed of an embedding layer, 2 LSTM layers, and a softmax layer. The vocabulary size is 10,000, and the LSTM hidden layer size is 200 for the small model, 650 for the medium model, and 1,500 for the large model. In the case of the large model, there are 15M parameters in the embedding layer, 18M parameters in each of the two LSTM layers, and 15M parameters in the softmax layer for a total of 66M parameters. Different hyperparameters are used to train the different-sized models. When pruning a model of a certain size, we use the same hyperparameters that were used for training the dense model of that size. We compare the performance of the dense models with sparse models pruned from medium and large to $80 \%$ , $8 5 \%$ , $90 \%$ , $9 5 \%$ , and $9 7 . 5 \%$ sparsity in Figure 4 and Table 3. In this case, we see that sparse models are able to outperform dense models which have significantly more parameters (note the log scale for the number of parameters). The $90 \%$ sparse large model (which has 6.6 million parameters and a perplexity of 80.24) is able to outperform the dense medium model (which has 19.8 million parameters and a perplexity of 83.37), a model which has 3 times more parameters. Compared with MobileNet, pruning PTB model likely gives better results because the PTB model is larger with significantly more parameters. Our results show that pruning works very well not only on the dense LSTM weights and dense softmax layer but also the dense embedding matrix. This suggests that during the optimization procedure the neural network can find a good sparse embedding for the words in the vocabulary that works well together with the sparse connectivity structure of the LSTM weights and softmax layer.
+
+![](images/3142a960213f198f84c7fd4421050ba3c88ee3b7fa5c9cd6c53ca287ab3b3125.jpg)  
+Figure 5: Comparison of sparse vs dense NMT models for English to German (EN-DE) and German to English (DE-EN) translation
+
+From Figure 4 and Table 3, we also see that the $85 \%$ sparse medium model (which has 3 million parameters and a perplexity of 85.17) outperforms the $9 5 \%$ sparse large model (which has 3.3 million parameters and a perplexity of 87.83). The accuracy of the $9 5 \%$ sparse large model is comparable to the accuracy of the $90 \%$ sparse medium model (which has 2 million parameters and a perplexity of 87.86). Together, these results suggest that there is an optimal compression range when pruning. In the case of PTB, pruning to $9 5 \%$ sparsity for a compression ratio of $2 0 \mathrm { x }$ significantly degrades the performance of the sparse model compared to pruning to $90 \%$ sparsity for a compression ratio of $1 0 \mathrm { x }$ , as seen in Figure 4 from the curve of perplexity vs. number of parameters traced by either of the sparse models. These results suggest that in order to get the best-performing sparse model of a certain size, we should train a dense model that is $5 \mathbf { X } { - } 1 0 \mathbf { X }$ larger and then prune to the desired number of parameters rather than taking the largest and best-performing dense model and pruning this model by $2 0 \mathrm { x }$ or more to the desired number of parameters, assuming that the difference in performance of the two dense baseline models is not that large. We note that it may be possible to obtain slightly better results for pruning to $9 5 \%$ sparsity or higher with more hyperparameter tuning, and the results we obtained for pruning a model of a certain size were from using exactly the same hyperparameter configuration as for training the dense model of that size.
+
+# 4.3 GOOGLE NEURAL MACHINE TRANSLATION
+
+The Google Neural Machine Translation (NMT) architecture is a seq2seq model with attention (Wu et al., 2016). We use the open-source TensorFlow implementation available at Luong et al. (2017). The model is based on an encoder-decoder architecture. The encoder has an embedding layer which maps the source vocabulary of 36,548 words into a $k$ -dimensional space, 1 bidirectional LSTM layer, and 3 standard LSTM layers. The decoder has an embedding layer which maps the target vocabulary of 36,548 words into a $k$ -dimensional space, 4 LSTM layers with attention, and finally a softmax layer. For the dense baseline model with number of units $k = 1 0 2 4$ , there are $3 7 . 4 \mathbf { M }$ parameters in each of the encoder embedding, decoder embedding, and softmax layers and $9 8 . 6 \mathbf { M }$ parameters in all of the LSTM layers for a total of 211M parameters. We apply pruning to all of the LSTM layers, embedding layers, and softmax layers, but we do not prune the attention parameters of which there are relatively few. The other dense models were obtained by varying the number of units $k$ . We use the WMT16 German and English dataset with news-test2013 as the dev set and news-test2015 as the test set. The BLEU score is reported as a measure of the translation quality. The learning rate schedule used for training the dense models is 170K iterations with initial learning rate 1.0 and 170K iterations with learning rate decay of 0.5 every 17K iterations. For pruning a dense model, the learning rate schedule we use is 70K iterations with initial learning rate 0.5 and 170K iterations with learning rate decay of 0.5 every 17K iterations, and all other hyperparameters were kept the same.
+
+Table 4: NMT sparse vs dense results   
+
+<table><tr><td># units</td><td>Sparsity</td><td> NNZ params</td><td>EN-DE BLEU score</td><td>DE-EN BLEU score</td></tr><tr><td>256</td><td>0%</td><td>34M</td><td>23.52</td><td>26.52</td></tr><tr><td>512</td><td>0%</td><td>81M</td><td>26.05</td><td>28.88</td></tr><tr><td>768</td><td>0%</td><td>140M</td><td>26.63</td><td>29.41</td></tr><tr><td>1024</td><td>0%</td><td>211M</td><td>26.77</td><td>29.47</td></tr><tr><td></td><td>80%</td><td>44M</td><td>26.86</td><td>29.50</td></tr><tr><td></td><td>85%</td><td>33M</td><td>26.52</td><td>29.24</td></tr><tr><td></td><td>90%</td><td>23M</td><td>26.19</td><td>28.81</td></tr></table>
+
+Since we noticed that the NMT training procedure had high variance, we tested several pruning schemes applied to NMT. Our standard implementation of gradual pruning increases the sparsity of every layer to the same sparsity level at each pruning step. We tested a variant which we call “layerwise constant” sparsity: instead of simultaneously increasing the sparsity of all layers to some sparsity level at each pruning step, we subdivide the pruning interval and increase the sparsity of one layer at a time to that sparsity level. This potentially has the effect of reducing the impact of pruning and allowing the network to recover better with training. Finally, we compared with “global” pruning: we prune the smallest magnitude weights across the entire network, regardless of which layer they are in. Global pruning produces a different sparsity level for each layer and was shown to perform well on NMT in the work of See et al. (2016). Overall, the layerwise constant pruning scheme performed best on average, so we report the results with the layerwise constant pruning scheme in Figure 5 and Table 4. We note that there is high variance in the results due to the stochasticity of the training process, as illustrated by the error bar in Figure 5 which is the standard deviation of the BLEU score of 10 randomly initialized and independently trained NMT models.
+
+The results in Table 4 show that for $80 \%$ sparsity (5x compression), the pruned model actually achieves a slightly higher BLEU score than the baseline model (though we note the error bar). For $8 5 \%$ sparsity, the BLEU score drops by around 0.25, and for $90 \%$ sparsity, the BLEU score drops by around 0.6. When we compare the performance of dense and sparse models in Figure 5 and Table 4, we again see that sparse models outperform even larger-sized dense models. The BLEU score of the dense model falls off quickly after $2 \mathbf { x }$ reduction in model size while the BLEU score of the sparse model starts to fall off only after ${ 5 } \mathbf { x }$ reduction in NNZ parameters. For example, the $90 \%$ sparse 1024-unit model is comparable to or outperforms the dense 512-unit model (26.19 vs 26.05 for EN-DE and 28.81 vs 28.88 for DE-EN) despite having $3 . 5 \mathrm { x }$ fewer NNZ params (23M vs 81M).
+
+# 5 DISCUSSION
+
+The net memory footprint of a sparse model includes the storage for the nonzero parameters and any auxiliary data structures needed for indexing these elements. Pruning models helps reduce the number of nonzero-valued connections in the network; however the overhead in sparse matrix storage inevitably diminishes the achievable compression ratio. The bit-mask sparse matrix representation requires 1 bit per matrix element indicating whether the element is nonzero, and a vector containing all the nonzero matrix elements. This representation incurs a constant overhead regardless of the model sparsity. In the compressed sparse row (column) storage (CSR(C)) adopted in Parashar et al. (2017), each nonzero parameter in the sparse matrix is associated with a count (usually stored as a 4 or 5 bit integer) of the number of zeros preceding it. The overhead in this case is proportional to the NNZ in the model. Table 5 compares these two representations for sparse-MobileNets. The CSR(C) representation can enable higher compression ratio for networks with high sparsity. Note, however, that the bit-mask representation offers marginally lower overhead at smaller sparsity levels.
+
+Table 5: Storage overheads associated with bit-mask and CSR(C) sparse matrix representations for sparse-MobileNets   
+
+<table><tr><td>Sparsity</td><td> NNZ params</td><td>Bit-mask (MB)</td><td>CSR(C) (MB)</td></tr><tr><td>0%</td><td>4.21M</td><td>N/A</td><td>N/A</td></tr><tr><td>50%</td><td>2.13M</td><td>0.52</td><td>1.06</td></tr><tr><td>75%</td><td>1.09M</td><td>0.52</td><td>0.54</td></tr><tr><td>90%</td><td>0.46M</td><td>0.52</td><td>0.23</td></tr><tr><td>95%</td><td>0.25M</td><td>0.52</td><td>0.13</td></tr></table>
+
+Table 6: Comparison of the performance of small-dense and large-sparse models. Model size calculations include overhead for sparse matrix storage and assumes 32-bit (4 bytes) per nonzero element.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">Small-dense</td><td colspan="2">Large-sparse</td></tr><tr><td>Model size (MB)</td><td>Accuracy (%)</td><td>Model size (MB)</td><td>Accuracy (%)</td></tr><tr><td rowspan="4">MobileNet</td><td>10.28</td><td>68.4</td><td>9.04</td><td>69.5</td></tr><tr><td>5.28</td><td>63.7</td><td>4.88</td><td>67.7</td></tr><tr><td>1.84</td><td>50.6</td><td>2.07</td><td>61.8</td></tr><tr><td></td><td></td><td>1.13</td><td>53.6</td></tr></table>
+
+In spite of this overhead, large-sparse models appear to achieve higher accuracy than small-dense models with comparable memory footprint. For instance, MobileNet with width multiplier 1 and sparsity $50 \%$ has similar footprint as MobileNet with width multiplier 0.75, but obtains higher accuracy. Table 6 further highlights the trade-off between model size and accuracy for dense and sparse models. The performance gap between large-sparse and small-dense models widens for larger models such as as the PTB language models and NMT (see Table 3 and Table 4). It is worth noting that the results presented in this work were obtained by training neural networks using 32-bit floating point representation. For neural networks trained to perform inference using reduced precision (8-bit integer, for instance) arithmetic, the memory overhead of sparse matrix storage represents a bigger fraction of the total memory footprint. Quantization of the parameters to a reduced precision number representation is also an effective method for model compression, and the interplay between model quantization and pruning and their collective impact on model accuracy merits a closer examination. We defer that investigation to a future extension to this work.
+
+# 6 CONCLUSION
+
+This work sheds light on the model size and accuracy trade-off encountered in pruned deep neural networks. We demonstrate that large-sparse models outperform comparably-sized small-dense models across a diverse set of neural network architectures. We also present a gradual pruning technique that can be applied with ease across these different architectures. We believe these results will encourage the adoption of model pruning as a tool for compressing neural networks for deployment in resource-constrained environments. At the same time, we hold the opinion that our results will provide further impetus to the hardware architecture community to customize the next generation of deep learning accelerator architectures to efficiently handle sparse matrix storage and computations.
+
+# REFERENCES
+
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+Jose M. Alvarez and Mathieu Salzmann. Learning the number of neurons in deep networks. CoRR, abs/1611.06321, 2016.   
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+Max Jaderberg, Andrea Vedaldi, and Andrew Zisserman. Speeding up convolutional neural networks with low rank expansions. CoRR, abs/1405.3866, 2014.   
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+Vadim Lebedev, Yaroslav Ganin, Maksim Rakhuba, Ivan V. Oseledets, and Victor S. Lempitsky. Speeding-up convolutional neural networks using fine-tuned cp-decomposition. CoRR, abs/1412.6553, 2014.   
+Yann LeCun, John S. Denker, and Sara A. Solla. Optimal brain damage. In D. S. Touretzky (ed.), Advances in Neural Information Processing Systems 2, pp. 598–605. Morgan-Kaufmann, 1990.   
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+Mohammad Rastegari, Vicente Ordonez, Joseph Redmon, and Ali Farhadi. Xnor-net: Imagenet classification using binary convolutional neural networks. CoRR, abs/1603.05279, 2016.   
+Abigail See, Minh-Thang Luong, and Christopher D. Manning. Compression of neural machine translation models via pruning. In CoNLL, pp. 291–301. ACL, 2016.   
+Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, and Zbigniew Wojna. Rethinking the inception architecture for computer vision. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2818–2826, 2016.   
+Vincent Vanhoucke, Andrew Senior, and Mark Z. Mao. Improving the speed of neural networks on cpus. In Deep Learning and Unsupervised Feature Learning Workshop, NIPS 2011, 2011.   
+Yonghui Wu, Mike Schuster, Zhifeng Chen, Quoc V. Le, Mohammad Norouzi, Wolfgang Macherey, Maxim Krikun, Yuan Cao, Qin Gao, Klaus Macherey, Jeff Klingner, Apurva Shah, Melvin Johnson, Xiaobing Liu, Lukasz Kaiser, Stephan Gouws, Yoshikiyo Kato, Taku Kudo, Hideto Kazawa, Keith Stevens, George Kurian, Nishant Patil, Wei Wang, Cliff Young, Jason Smith, Jason Riesa, Alex Rudnick, Oriol Vinyals, Greg Corrado, Macduff Hughes, and Jeffrey Dean. Google’s neural machine translation system: Bridging the gap between human and machine translation. CoRR, abs/1609.08144, 2016.   
+Wojciech Zaremba, Ilya Sutskever, and Oriol Vinyals. Recurrent neural network regularization. CoRR, abs/1409.2329, 2014.   
+Shijin Zhang, Zidong Du, Lei Zhang, Huiying Lan, Shaoli Liu, Ling Li, Qi Guo, Tianshi Chen, and Yunji Chen. Cambricon-x: An accelerator for sparse neural networks. In Microarchitecture (MICRO), 2016 49th Annual IEEE/ACM International Symposium on, pp. 1–12. IEEE, 2016.   
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+    {
+        "type": "text",
+        "text": "TO PRUNE, OR NOT TO PRUNE: EXPLORING THE EFFICACY OF PRUNING FOR MODEL COMPRESSION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
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+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Model pruning seeks to induce sparsity in a deep neural network’s various connection matrices, thereby reducing the number of nonzero-valued parameters in the model. Recent reports (Han et al., 2015a; Narang et al., 2017) prune deep networks at the cost of only a marginal loss in accuracy and achieve a sizable reduction in model size. This hints at the possibility that the baseline models in these experiments are perhaps severely over-parameterized at the outset and a viable alternative for model compression might be to simply reduce the number of hidden units while maintaining the model’s dense connection structure, exposing a similar trade-off in model size and accuracy. We investigate these two distinct paths for model compression within the context of energy-efficient inference in resource-constrained environments and propose a new gradual pruning technique that is simple and straightforward to apply across a variety of models/datasets with minimal tuning and can be seamlessly incorporated within the training process. We compare the accuracy of large, but pruned models (large-sparse) and their smaller, but dense (small-dense) counterparts with identical memory footprint. Across a broad range of neural network architectures (deep CNNs, stacked LSTM, and seq2seq LSTM models), we find large-sparse models to consistently outperform small-dense models and achieve up to $1 0 \\mathrm { x }$ reduction in number of non-zero parameters with minimal loss in accuracy. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Over the past few years, deep neural networks have achieved state-of-the-art performance on several challenging tasks in the domains of computer vision, speech recognition, and natural language processing. Driven by increasing amounts of data and computational power, deep learning models have become bigger and deeper to better learn from data. While these models are typically deployed in a datacenter back-end, preserving user privacy and reducing user-perceived query times mandate the migration of the intelligence offered by these deep neural networks towards edge computing devices. Deploying large, accurate deep learning models to resource-constrained computing environments such as mobile phones, smart cameras etc. for on-device inference poses a few key challenges. Firstly, state-of-the-art deep learning models routinely have millions of parameters requiring \\~MBs of storage, whereas on-device memory is limited. Furthermore, it is not uncommon for even a single model inference to invoke \\~billions of memory accesses and arithmetic operations, all of which consume power and dissipate heat which may drain the limited battery capacity and/or test the device’s thermal limits. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Confronting these challenges, a growing body of work has emerged that intends to discover methods for compressing neural network models while limiting any potential loss in model quality. Latencysensitive workloads relying on energy-efficient on-device neural network inference are often memory bandwidth-bound, and model compression offers the two-fold benefit of reducing the total number of energy-intensive memory accesses as well as improving the inference time due to an effectively higher memory bandwidth for fetching compressed model parameters. Within the realm of model compression techniques, pruning away (forcing to zero) the less salient connections (parameters) in the neural network has been shown to reduce the number of nonzero parameters in the model with little to no loss in the final model quality. Model pruning enables trading off a small degradation in model quality for a reduction in model size, potentially reaping improvements in inference time and energy-efficiency. The resulting pruned model typically has sparse connection matrices, so efficient inference using these sparse models requires purpose-built hardware capable of loading sparse matrices and/or performing sparse matrix-vector operations (Zhang et al., 2016; Han et al., 2016; Parashar et al., 2017). Also, representing sparse matrices carries with it an additional storage overhead increasing the model’s net memory footprint which must also be taken into consideration. ",
+        "bbox": [
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "In this work, we perform a closer examination of the effectiveness of model pruning as a means for model compression. From the perspective of on-device neural network inference, given a bound on the model’s memory footprint, how can we arrive at the most accurate model? We aim to answer this question by comparing the quality of the models obtained through two distinct methods: (1) training a large model, but pruned to obtain a sparse model with a small number of nonzero parameters (large-sparse); and (2) training a small-dense model with size comparable to the large-sparse model. Both of these methods expose a model accuracy and size tradeoff, but differ remarkably in terms of their implications on the design of the underlying hardware architecture. For this comparative study, we pick models across a diverse set of application domains: InceptionV3 (Szegedy et al., 2016) and MobileNets (Howard et al., 2017) for image recognitions tasks, stacked LSTMs for language modeling, and seq2seq models used in Google’s Neural Machine Translation (Wu et al., 2016) system. In the process of this investigation, we also develop a simple gradual pruning approach that requires minimal tuning and can be seamlessly incorporated within the training process and demonstrate its applicability and performance on an assortment of neural network architectures. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            387,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Early works in the 1990s (LeCun et al., 1990; Hassibi et al., 1993) performed pruning using a second-order Taylor approximation of the increase in the loss function of the network when a weight is set to zero. In Optimal Brain Damage (LeCun et al., 1990), the saliency for each weight was computed using a diagonal Hessian approximation, and the low-saliency weights were pruned from the network and the network was retrained. In Optimal Brain Surgeon (Hassibi et al., 1993), the saliency for each weight was computed using the inverse Hessian matrix, and the low-saliency weights were pruned and all other weights in the network were updated using the Hessian matrix. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "More recently, magnitude-based weight pruning methods have become popular techniques for network pruning (Han et al., 2015b;a; See et al., 2016; Narang et al., 2017). Magnitude-based weight pruning techniques are computationally efficient, scaling to large networks and datasets. Our automated gradual pruning algorithm prunes the smallest magnitude weights to achieve a preset level of network sparsity. In contrast with the works listed above, our paper focuses on comparing the model accuracy and size tradeoff of large-sparse versus small-dense models. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A work similar to ours is the work by Narang et al. (2017) on pruning a RNN and GRU model for speech recognition and showing that a sparse RNN that was pruned outperformed a dense RNN trained normally of comparable size. While they provide one data point comparing the performance of a sparse vs dense model, our work does an extensive comparison of sparse vs dense models across a wide range of models in different domains (vision and NLP). Narang et al. also introduce a gradual pruning scheme based on pruning all the weights in a layer less than some threshold (manually chosen) which is linear with some slope in phase 1 and linear with some slope in phase 2 followed by normal training. Compared to their approach, we do not have two phases and do not have to choose two slopes, and we do not need to choose weight thresholds for each layer (we rely on a sparsity schedule which determines the weight thresholds). Thus, our technique is simpler, doesn’t require much hyperparameter tuning, and is shown to perform well across different models. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Within the context of reducing model size by removing redundant connections, several recent works (Anwar et al., 2015; Lebedev & Lempitsky, 2015; Li et al., 2016; Changpinyo et al., 2017) propose techniques to prune and induce sparsity in a structured way, motivated primarily by the desire to speedup computations on existing hardware architectures optimized for dense linear algebra. Such techniques perform coarse-grain pruning and depend critically on the structure of the convolutional layers, and may not be directly extensible to other neural network architectures that lack such structural properties (LSTMs for instance). On the contrary, our method does not make any assumptions about the structure of the network or its constituent layers and is therefore more generally applicable. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "While pruning focuses on reducing the number of non-zero parameters, in principle, model pruning can be used in conjunction with other techniques to further reduce model size. Quantization techniques aim to reduce the number of bits required to represent each parameter from 32-bit floats to 8 bits or fewer. Different quantization techniques such as fixed-point quantization (Vanhoucke et al., 2011) or vector quantization (Gong et al., 2014) achieve different compression ratios and accuracies but also require different software or hardware to support inference at runtime. Pruning can be combined with quantization to achieve maximal compression (Han et al., 2015a). In addition, an emerging area of research is low precision networks where the parameters and/or activations are quantized to 4 bits or fewer (Courbariaux et al., 2015; Lin et al., 2015; Hubara et al., 2016; Rastegari et al., 2016; Zhu et al., 2016). Besides quantization, other potentially complementary approaches to reducing model size include low-rank matrix factorization (Denil et al., 2013; Denton et al., 2014; Jaderberg et al., 2014; Lebedev et al., 2014) and group sparsity regularization to arrive at an optimal layer size (Alvarez & Salzmann, 2016). ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/b8a464b65ac8989f1c3928533cfcfc3b5ca03050aa33b8deb596993adf0875b5.jpg",
+        "image_caption": [
+            "Figure 1: Sparsity function used for gradual pruning "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/f82e602a2923e1425c1b99d3a5e287ad10dde9b42504217a0f622a8a318f05c7.jpg",
+        "table_caption": [
+            "Table 1: Model size and accuracy tradeoff for sparse-InceptionV3 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Sparsity</td><td>NNZ params</td><td>Top-1 acc.</td><td>Top-5 acc.</td></tr><tr><td>0%</td><td>27.1M</td><td>78.1%</td><td>94.3%</td></tr><tr><td>50%</td><td>13.6M</td><td>78.0%</td><td>94.2%</td></tr><tr><td>75%</td><td>6.8M</td><td>76.1%</td><td>93.2%</td></tr><tr><td>87.5%</td><td>3.3M</td><td>74.6%</td><td>92.5%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "3 METHODS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We extend the TensorFlow (Abadi et al., 2015) framework to prune the network’s connections during training. For every layer chosen to be pruned, we add a binary mask variable which is of the same size and shape as the layer’s weight tensor and determines which of the weights participate in the forward execution of the graph. We inject ops into the TensorFlow training graph to sort the weights in that layer by their absolute values and mask to zero the smallest magnitude weights until some desired sparsity level $s$ is reached. The back-propagated gradients flow through the binary masks, and the weights that were masked in the forward execution do not get updated in the backpropagation step. We introduce a new automated gradual pruning algorithm in which the sparsity is increased from an initial sparsity value $s _ { i }$ (usually 0) to a final sparsity value $s _ { f }$ over a span of $n$ pruning steps, starting at training step $t _ { 0 }$ and with pruning frequency $\\Delta t$ : ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a614fbae9855ab65d6d851147fab0f74cd02540d1a6bff2625ad2d58fe0bf2c1.jpg",
+        "text": "$$\ns _ { t } = s _ { f } + ( s _ { i } - s _ { f } ) \\left( 1 - \\frac { t - t _ { 0 } } { n \\Delta t } \\right) ^ { 3 } \\mathrm { ~ f o r ~ } t \\in \\{ t _ { 0 } , ~ t _ { 0 } + \\Delta t , ~ . . . , ~ t _ { 0 } + n \\Delta t \\}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "The binary weight masks are updated every $\\Delta t$ steps as the network is trained to gradually increase the sparsity of the network while allowing the network training steps to recover from any pruninginduced loss in accuracy. In our experience, varying the pruning frequency $\\Delta t$ between 100 and 1000 training steps had a negligible impact on the final model quality. Once the model achieves the target sparsity $s _ { f }$ , the weight masks are no longer updated. The intuition behind this sparsity function in equation (1) is to prune the network rapidly in the initial phase when the redundant connections are abundant and gradually reduce the number of weights being pruned each time as there are fewer and fewer weights remaining in the network, as illustrated in Figure 1. In the experimental results presented in this paper, pruning is initiated after the model has been trained for a few epochs or from a pre-trained model. This determines the value for the hyperparameter $t _ { 0 }$ . A suitable choice for $n$ is largely dependent on the learning rate schedule. Stochastic gradient descent (and its many variants) typically decay the learning rate during training, and we have observed that pruning in the presence of an exceedingly small learning rate makes it difficult for the subsequent training steps to recover from the loss in accuracy caused by forcing the weights to zero. At the same time, pruning with too high of a learning rate may mean pruning weights when the weights have not yet converged to a good solution, so the pruning schedule should be chosen closely with the learning rate schedule. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/7fbebfe6d2a1ec0deba98179b35da1400747e9503979eb08eb7c1464ff47b4e3.jpg",
+        "image_caption": [
+            "Figure 2: (a) The gradual sparsity function and exponentially decaying learning rate used for training sparse-InceptionV3 models. (b) Evolution of the model’s accuracy during the training process "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "img_path": "images/8e5c10d27205284de0780dc0e30b8cb8150f92a11177d9cc85416f827808d715.jpg",
+        "image_caption": [
+            "Figure 3: MobileNet sparse vs dense results "
+        ],
+        "image_footnote": [],
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+    {
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+        "img_path": "images/702b771ad2805dc43f0e503ff094d90992d665a3ee98080cd91e746fe237d7a2.jpg",
+        "image_caption": [
+            "Figure 4: PTB sparse vs dense results "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "Figure 2a shows the learning rate and the pruning schedule used for training sparse-InceptionV3 (Szegedy et al., 2016) models. All the convolutional layers in this model are pruned using the same sparsity function, and pruning occurs in the regime where the learning rate is still reasonably high to allow the network to heal from the pruning-induced damage. Figure 2b offers more insight into how this pruning scheme interacts with the training procedure. For the $8 7 . 5 \\%$ sparse model, with the gradual increase in sparsity, there comes a point when the model suffers a near-catastrophic degradation, but recovers nearly just as quickly with continued training. This behavior is more pronounced in the models trained to have higher sparsity. Table 1 compares the performance of sparse-InceptionV3 models pruned to varying extents. As expected, there is a gradual degradation in the model quality as the sparsity increases. However, a $50 \\%$ sparse model performs just as well as the baseline $0 \\%$ sparsity), and there is only a $2 \\%$ decrease in top-5 classification accuracy for the $8 7 . 5 \\%$ sparse model which offers an ${ 8 } \\mathbf { { x } }$ reduction in number of nonzero (NNZ) model parameters. Also note that since the weights are initialized randomly, the sparsity in the weight tensors does not exhibit any specific structure. Furthermore, the pruning method described here does not depend on any specific property of the network or the constituent layers, and can be extended directly to a wide-range of neural network architectures. ",
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+    },
+    {
+        "type": "text",
+        "text": "4 COMPARING large-sparse AND small-dense MODELS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.1 MOBILENETS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "MobileNets are a class of efficient convolutional neural networks designed specifically for mobile vision applications (Howard et al., 2017). Instead of using standard convolutions, MobileNets are based on a form of factorized convolutions called depthwise separable convolution. Depthwise separable convolutions consist of a depthwise convolution followed by a 1x1 convolution called a pointwise convolution. This factorization significantly reduces the number of parameters in the model by filtering and combining input channels in two separate steps instead of together as in the standard convolution. The MobileNet architecture consists of one standard convolution layer acting on the input image, a stack of depthwise separable convolutions, and finally averaging pooling and fully connected layers. For the dense baseline model with width multiplier 1.0, there are a total of 4.21M parameters, $9 9 \\%$ of which are in the 1x1 pointwise convolution layers $( 7 4 . 6 \\% )$ and fully connected layers $( 2 4 . 3 \\% )$ . We do not prune the parameters in the one standard convolution layer and in the depthwise convolution layers since there are very few parameters in those layers $( 1 . 1 \\% )$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/d013f371dad74e7468a7894d9dc64bc6a32ec487634232bea56ecfc25b9370ba.jpg",
+        "table_caption": [
+            "Table 2: MobileNet sparse vs dense results "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Width</td><td>Sparsity</td><td>NNZ params</td><td>Top-1 acc.</td><td>Top-5 acc.</td></tr><tr><td>0.25</td><td>0%</td><td>0.46M</td><td>50.6%</td><td>75.0%</td></tr><tr><td>0.5</td><td>0%</td><td>1.32M</td><td>63.7%</td><td>85.4%</td></tr><tr><td>0.75</td><td>0%</td><td>2.57M</td><td>68.4%</td><td>88.2%</td></tr><tr><td rowspan=\"4\">1.0</td><td>0%</td><td>4.21M</td><td>70.6%</td><td>89.5%</td></tr><tr><td>50%</td><td>2.13M</td><td>69.5%</td><td>89.5%</td></tr><tr><td>75%</td><td>1.09M</td><td>67.7%</td><td>88.5%</td></tr><tr><td>90%</td><td>0.46M</td><td>61.8%</td><td>84.7%</td></tr><tr><td></td><td>95%</td><td>0.25M</td><td>53.6%</td><td>78.9%</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/77556397e36ddd1724721f575189b3ec5ef1727e7972b615015764bf1708cbb1.jpg",
+        "table_caption": [
+            "Table 3: PTB sparse vs dense results "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Sparsity</td><td>NNZ params</td><td>Per- plexity</td></tr><tr><td>Small</td><td>0%</td><td>4.6M</td><td>115.30</td></tr><tr><td rowspan=\"5\">Medium</td><td>0%</td><td>19.8M</td><td>83.37</td></tr><tr><td>80%</td><td>4.0M</td><td>83.87</td></tr><tr><td>85%</td><td>3.0M</td><td>85.17</td></tr><tr><td>90%</td><td>2.0M</td><td>87.86</td></tr><tr><td>95% 97.5%</td><td>1.0M 0.5M</td><td>96.30 113.6</td></tr><tr><td rowspan=\"5\">Large</td><td>0%</td><td>66M</td><td>78.45</td></tr><tr><td>80%</td><td>13.2M</td><td>77.52</td></tr><tr><td>85%</td><td>9.9M</td><td>78.31</td></tr><tr><td>90%</td><td>6.6M</td><td>80.24</td></tr><tr><td>95%</td><td>3.3M</td><td>87.83</td></tr><tr><td></td><td>97.5%</td><td>1.7M</td><td>103.20</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The width multiplier is a parameter of the MobileNet network that allows trading off the accuracy of the model with the number of parameters and computational cost. The width multiplier of the baseline model is 1.0. For a given width multiplier $\\alpha \\in ( 0 , 1 ]$ , the number of input channels and the number of output channels in each layer is scaled by $\\alpha$ relative to the baseline 1.0 model. We compare the performance of dense MobileNets trained with width multipliers 0.75, 0.5, and 0.25 with the performance of sparse MobileNets pruned from dense 1.0 MobileNet in Figure 3 and Table 2 on the ImageNet dataset. We see that for a given number of non-zero parameters, sparse MobileNets are able to outperform dense MobileNets. For example, the $7 5 \\%$ sparse model (which has 1.09 million parameters and a top-1 accuracy of $6 7 . 7 \\%$ ) outperforms the dense 0.5 MobileNet (which has 1.32 million parameters and a top-1 accuracy of $6 3 . 7 \\%$ ) by $4 \\%$ in top-1 accuracy while being smaller. Similarly, the $90 \\%$ sparse model (which has 0.46 million parameters and a top-1 accuracy of $6 1 . 8 \\%$ ) outperforms the dense 0.25 MobileNet (which has 0.46 million parameters and a top-1 accuracy of $5 0 . 6 \\%$ by $1 0 . 2 \\%$ in top-1 accuracy while having the same number of non-zero parameters. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Overall, pruning is a promising approach for model compression even for an architecture that was designed to be compact and efficient by using depthwise separable convolutions instead of standard convolutions as a factorization-like technique to reduce the number of parameters. The sparsity parameter is shown to be an effective way to trade off the accuracy of a model with its memory usage and compares favorably with the width multiplier in MobileNet. Training a sparse MobileNet using our gradual pruning algorithm is also easy. For pruning a dense MobileNet, we used the same learning rate schedule as for training a dense MobileNet but with an initial learning rate 10 times smaller than for training a dense MobileNet, and all other hyperparameters were kept the same. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 PENN TREE BANK (PTB) LANGUAGE MODEL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            854,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We train an LSTM language model on the Penn Tree Bank dataset using the models and training procedure described in Zaremba et al. (2014). At each time step, the LSTM language model outputs the probability of the next word in the sentence given the history of previous words. The loss function is the average negative log probability of the target words, and the perplexity is the exponential of the loss function. The language model is composed of an embedding layer, 2 LSTM layers, and a softmax layer. The vocabulary size is 10,000, and the LSTM hidden layer size is 200 for the small model, 650 for the medium model, and 1,500 for the large model. In the case of the large model, there are 15M parameters in the embedding layer, 18M parameters in each of the two LSTM layers, and 15M parameters in the softmax layer for a total of 66M parameters. Different hyperparameters are used to train the different-sized models. When pruning a model of a certain size, we use the same hyperparameters that were used for training the dense model of that size. We compare the performance of the dense models with sparse models pruned from medium and large to $80 \\%$ , $8 5 \\%$ , $90 \\%$ , $9 5 \\%$ , and $9 7 . 5 \\%$ sparsity in Figure 4 and Table 3. In this case, we see that sparse models are able to outperform dense models which have significantly more parameters (note the log scale for the number of parameters). The $90 \\%$ sparse large model (which has 6.6 million parameters and a perplexity of 80.24) is able to outperform the dense medium model (which has 19.8 million parameters and a perplexity of 83.37), a model which has 3 times more parameters. Compared with MobileNet, pruning PTB model likely gives better results because the PTB model is larger with significantly more parameters. Our results show that pruning works very well not only on the dense LSTM weights and dense softmax layer but also the dense embedding matrix. This suggests that during the optimization procedure the neural network can find a good sparse embedding for the words in the vocabulary that works well together with the sparse connectivity structure of the LSTM weights and softmax layer. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/3142a960213f198f84c7fd4421050ba3c88ee3b7fa5c9cd6c53ca287ab3b3125.jpg",
+        "image_caption": [
+            "Figure 5: Comparison of sparse vs dense NMT models for English to German (EN-DE) and German to English (DE-EN) translation "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
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+            253
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "From Figure 4 and Table 3, we also see that the $85 \\%$ sparse medium model (which has 3 million parameters and a perplexity of 85.17) outperforms the $9 5 \\%$ sparse large model (which has 3.3 million parameters and a perplexity of 87.83). The accuracy of the $9 5 \\%$ sparse large model is comparable to the accuracy of the $90 \\%$ sparse medium model (which has 2 million parameters and a perplexity of 87.86). Together, these results suggest that there is an optimal compression range when pruning. In the case of PTB, pruning to $9 5 \\%$ sparsity for a compression ratio of $2 0 \\mathrm { x }$ significantly degrades the performance of the sparse model compared to pruning to $90 \\%$ sparsity for a compression ratio of $1 0 \\mathrm { x }$ , as seen in Figure 4 from the curve of perplexity vs. number of parameters traced by either of the sparse models. These results suggest that in order to get the best-performing sparse model of a certain size, we should train a dense model that is $5 \\mathbf { X } { - } 1 0 \\mathbf { X }$ larger and then prune to the desired number of parameters rather than taking the largest and best-performing dense model and pruning this model by $2 0 \\mathrm { x }$ or more to the desired number of parameters, assuming that the difference in performance of the two dense baseline models is not that large. We note that it may be possible to obtain slightly better results for pruning to $9 5 \\%$ sparsity or higher with more hyperparameter tuning, and the results we obtained for pruning a model of a certain size were from using exactly the same hyperparameter configuration as for training the dense model of that size. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 GOOGLE NEURAL MACHINE TRANSLATION",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The Google Neural Machine Translation (NMT) architecture is a seq2seq model with attention (Wu et al., 2016). We use the open-source TensorFlow implementation available at Luong et al. (2017). The model is based on an encoder-decoder architecture. The encoder has an embedding layer which maps the source vocabulary of 36,548 words into a $k$ -dimensional space, 1 bidirectional LSTM layer, and 3 standard LSTM layers. The decoder has an embedding layer which maps the target vocabulary of 36,548 words into a $k$ -dimensional space, 4 LSTM layers with attention, and finally a softmax layer. For the dense baseline model with number of units $k = 1 0 2 4$ , there are $3 7 . 4 \\mathbf { M }$ parameters in each of the encoder embedding, decoder embedding, and softmax layers and $9 8 . 6 \\mathbf { M }$ parameters in all of the LSTM layers for a total of 211M parameters. We apply pruning to all of the LSTM layers, embedding layers, and softmax layers, but we do not prune the attention parameters of which there are relatively few. The other dense models were obtained by varying the number of units $k$ . We use the WMT16 German and English dataset with news-test2013 as the dev set and news-test2015 as the test set. The BLEU score is reported as a measure of the translation quality. The learning rate schedule used for training the dense models is 170K iterations with initial learning rate 1.0 and 170K iterations with learning rate decay of 0.5 every 17K iterations. For pruning a dense model, the learning rate schedule we use is 70K iterations with initial learning rate 0.5 and 170K iterations with learning rate decay of 0.5 every 17K iterations, and all other hyperparameters were kept the same. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/40b602336ae5ebdf41e128cacf50534402f00f80f749fc5d68752a9377b7d9e3.jpg",
+        "table_caption": [
+            "Table 4: NMT sparse vs dense results "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td># units</td><td>Sparsity</td><td> NNZ params</td><td>EN-DE BLEU score</td><td>DE-EN BLEU score</td></tr><tr><td>256</td><td>0%</td><td>34M</td><td>23.52</td><td>26.52</td></tr><tr><td>512</td><td>0%</td><td>81M</td><td>26.05</td><td>28.88</td></tr><tr><td>768</td><td>0%</td><td>140M</td><td>26.63</td><td>29.41</td></tr><tr><td>1024</td><td>0%</td><td>211M</td><td>26.77</td><td>29.47</td></tr><tr><td></td><td>80%</td><td>44M</td><td>26.86</td><td>29.50</td></tr><tr><td></td><td>85%</td><td>33M</td><td>26.52</td><td>29.24</td></tr><tr><td></td><td>90%</td><td>23M</td><td>26.19</td><td>28.81</td></tr></table>",
+        "bbox": [
+            223,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Since we noticed that the NMT training procedure had high variance, we tested several pruning schemes applied to NMT. Our standard implementation of gradual pruning increases the sparsity of every layer to the same sparsity level at each pruning step. We tested a variant which we call “layerwise constant” sparsity: instead of simultaneously increasing the sparsity of all layers to some sparsity level at each pruning step, we subdivide the pruning interval and increase the sparsity of one layer at a time to that sparsity level. This potentially has the effect of reducing the impact of pruning and allowing the network to recover better with training. Finally, we compared with “global” pruning: we prune the smallest magnitude weights across the entire network, regardless of which layer they are in. Global pruning produces a different sparsity level for each layer and was shown to perform well on NMT in the work of See et al. (2016). Overall, the layerwise constant pruning scheme performed best on average, so we report the results with the layerwise constant pruning scheme in Figure 5 and Table 4. We note that there is high variance in the results due to the stochasticity of the training process, as illustrated by the error bar in Figure 5 which is the standard deviation of the BLEU score of 10 randomly initialized and independently trained NMT models. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The results in Table 4 show that for $80 \\%$ sparsity (5x compression), the pruned model actually achieves a slightly higher BLEU score than the baseline model (though we note the error bar). For $8 5 \\%$ sparsity, the BLEU score drops by around 0.25, and for $90 \\%$ sparsity, the BLEU score drops by around 0.6. When we compare the performance of dense and sparse models in Figure 5 and Table 4, we again see that sparse models outperform even larger-sized dense models. The BLEU score of the dense model falls off quickly after $2 \\mathbf { x }$ reduction in model size while the BLEU score of the sparse model starts to fall off only after ${ 5 } \\mathbf { x }$ reduction in NNZ parameters. For example, the $90 \\%$ sparse 1024-unit model is comparable to or outperforms the dense 512-unit model (26.19 vs 26.05 for EN-DE and 28.81 vs 28.88 for DE-EN) despite having $3 . 5 \\mathrm { x }$ fewer NNZ params (23M vs 81M). ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The net memory footprint of a sparse model includes the storage for the nonzero parameters and any auxiliary data structures needed for indexing these elements. Pruning models helps reduce the number of nonzero-valued connections in the network; however the overhead in sparse matrix storage inevitably diminishes the achievable compression ratio. The bit-mask sparse matrix representation requires 1 bit per matrix element indicating whether the element is nonzero, and a vector containing all the nonzero matrix elements. This representation incurs a constant overhead regardless of the model sparsity. In the compressed sparse row (column) storage (CSR(C)) adopted in Parashar et al. (2017), each nonzero parameter in the sparse matrix is associated with a count (usually stored as a 4 or 5 bit integer) of the number of zeros preceding it. The overhead in this case is proportional to the NNZ in the model. Table 5 compares these two representations for sparse-MobileNets. The CSR(C) representation can enable higher compression ratio for networks with high sparsity. Note, however, that the bit-mask representation offers marginally lower overhead at smaller sparsity levels. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/02829d3b815c8caa37389551d79ec640eed8d6c6d770ffc730b859e791ee323d.jpg",
+        "table_caption": [
+            "Table 5: Storage overheads associated with bit-mask and CSR(C) sparse matrix representations for sparse-MobileNets "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Sparsity</td><td> NNZ params</td><td>Bit-mask (MB)</td><td>CSR(C) (MB)</td></tr><tr><td>0%</td><td>4.21M</td><td>N/A</td><td>N/A</td></tr><tr><td>50%</td><td>2.13M</td><td>0.52</td><td>1.06</td></tr><tr><td>75%</td><td>1.09M</td><td>0.52</td><td>0.54</td></tr><tr><td>90%</td><td>0.46M</td><td>0.52</td><td>0.23</td></tr><tr><td>95%</td><td>0.25M</td><td>0.52</td><td>0.13</td></tr></table>",
+        "bbox": [
+            290,
+            141,
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+            241
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/d0ae11a0f96e1e32f47ff0aa1f01667127a060e24a90ad6199fe98088e9a7106.jpg",
+        "table_caption": [
+            "Table 6: Comparison of the performance of small-dense and large-sparse models. Model size calculations include overhead for sparse matrix storage and assumes 32-bit (4 bytes) per nonzero element. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"2\">Small-dense</td><td colspan=\"2\">Large-sparse</td></tr><tr><td>Model size (MB)</td><td>Accuracy (%)</td><td>Model size (MB)</td><td>Accuracy (%)</td></tr><tr><td rowspan=\"4\">MobileNet</td><td>10.28</td><td>68.4</td><td>9.04</td><td>69.5</td></tr><tr><td>5.28</td><td>63.7</td><td>4.88</td><td>67.7</td></tr><tr><td>1.84</td><td>50.6</td><td>2.07</td><td>61.8</td></tr><tr><td></td><td></td><td>1.13</td><td>53.6</td></tr></table>",
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+        "text": "In spite of this overhead, large-sparse models appear to achieve higher accuracy than small-dense models with comparable memory footprint. For instance, MobileNet with width multiplier 1 and sparsity $50 \\%$ has similar footprint as MobileNet with width multiplier 0.75, but obtains higher accuracy. Table 6 further highlights the trade-off between model size and accuracy for dense and sparse models. The performance gap between large-sparse and small-dense models widens for larger models such as as the PTB language models and NMT (see Table 3 and Table 4). It is worth noting that the results presented in this work were obtained by training neural networks using 32-bit floating point representation. For neural networks trained to perform inference using reduced precision (8-bit integer, for instance) arithmetic, the memory overhead of sparse matrix storage represents a bigger fraction of the total memory footprint. Quantization of the parameters to a reduced precision number representation is also an effective method for model compression, and the interplay between model quantization and pruning and their collective impact on model accuracy merits a closer examination. We defer that investigation to a future extension to this work. ",
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+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
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+        "text": "This work sheds light on the model size and accuracy trade-off encountered in pruned deep neural networks. We demonstrate that large-sparse models outperform comparably-sized small-dense models across a diverse set of neural network architectures. We also present a gradual pruning technique that can be applied with ease across these different architectures. We believe these results will encourage the adoption of model pruning as a tool for compressing neural networks for deployment in resource-constrained environments. At the same time, we hold the opinion that our results will provide further impetus to the hardware architecture community to customize the next generation of deep learning accelerator architectures to efficiently handle sparse matrix storage and computations. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+        "page_idx": 8
+    },
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+        "type": "text",
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parse/train/SJgNwi09Km/SJgNwi09Km.md

@@ -0,0 +1,334 @@
+# LEARNING LATENT SUPERSTRUCTURES IN VARIATIONAL AUTOENCODERS FOR DEEP MULTIDIMENSIONAL CLUSTERING
+
+Xiaopeng $\mathbf { L i } ^ { 1 }$ , Zhourong Chen1, Leonard K. M. Poon2 and Nevin L. Zhang1
+
+1 Department of Computer Science and Engineering The Hong Kong University of Science and Technology 2 Department of Mathematics & Information Technology The Education University of Hong Kong {xlibo,zchenbb,lzhang}@cse.ust.hk, kmpoon@eduhk.hk
+
+# ABSTRACT
+
+We investigate a variant of variational autoencoders where there is a superstructure of discrete latent variables on top of the latent features. In general, our superstructure is a tree structure of multiple super latent variables and it is automatically learned from data. When there is only one latent variable in the superstructure, our model reduces to one that assumes the latent features to be generated from a Gaussian mixture model. We call our model the latent tree variational autoencoder (LTVAE). Whereas previous deep learning methods for clustering produce only one partition of data, LTVAE produces multiple partitions of data, each being given by one super latent variable. This is desirable because high dimensional data usually have many different natural facets and can be meaningfully partitioned in multiple ways.
+
+# 1 INTRODUCTION
+
+Clustering is a fundamental task in unsupervised machine learning, and it is central to many datadriven application domains. Cluster analysis partitions all the data into disjoint groups, and one can understand the structure of the data by examining examples in each group. Many clustering methods have been proposed in the literature (Aggarwal & Reddy, 2013), such as $k$ -means (MacQueen et al., 1967), Gaussian mixture models (Christopher, 2016) and spectral clustering (Von Luxburg, 2007). Conventional clustering methods are generally applied directly on the original data space. However, it is challenging to perform cluster analysis on high dimensional and unstructured data (Steinbach et al., 2004), such as images. It is not only because the dimensionality is high, but also because the original data space is too complex to interpret, e.g. there are semantic gaps between pixel values and objects in images.
+
+Recently, deep learning based clustering methods have been proposed that simultanously learn nonlinear embeddings through deep neural networks and perform cluster analysis on the embedding space. The representation learning process learns effective high-level representations from high dimensional data and helps the cluster analysis. This is typically achieved by unsupervised deep learning methods, such as restricted Boltzmann machine (RBM) (Hinton et al., 2006; Hinton & Salakhutdinov, 2006), autoencoders (AE) (Vincent et al., 2008; 2010), variational autoencoders (VAE) (Kingma & Welling, 2014), etc. Previous deep learning based clustering methods (Xie et al., 2016; Guo et al., 2017; Jiang et al., 2017; Yang et al., 2017) assume one single partition over the data and that all attributes define that partition. In real-world applications, however, the assumptions are usually not true. High-dimensional data are often multifaceted and can be meaningfully partitioned in multiple ways based on subsets of attributes (Chen et al., 2012). For example, a student population can be clustered in one way based on course grades and in another way based on extracurricular activities. Movie reviews can be clustered based on both sentiment (positive or negative) and genre (comedy, action, war, etc.). It is challenging to discover the multi-facet structures of data, especially for high-dimensional data.
+
+To resolve the above issues, we propose an unsupervised learning method, latent tree variational autoencoder (LTVAE) to learn latent superstructures in variational autoencoders, and simultaneously perform representation learning and structure learning. LTVAE is a generative model, where the data is assumed to be generated from latent features through neural networks, while the latent features themselves are generated from tree-structured Bayesian networks with another level of latent variables as shown in Fig. 1. Each of those latent variables defines a facet of clustering. The proposed method automatically selects subsets of latent features for each facet, and learns the dependency structure among different facets. This is achieved through systematic structure learning. Consequently, LTVAE is able to discover complex structures of data rather than one partition. We also propose efficient learning algorithms for LTVAE with gradient descent and Stepwise EM through message passing.
+
+The rest of the paper is organized as follows. The related works are reviewed in Section 2. We introduce the proposed method and learning algorithms in Section 3. In Section 4, we present the empirical results. The conclusion is given in Section 5.
+
+# 2 RELATED WORKS
+
+Clustering has been extensively studied in the literature in many aspects (Aggarwal & Reddy, 2013). More complex clustering methods related to structure learning using Bayesian nonparametrics have been proposed, like Dirichlet Process (Blei et al., 2006), Hierarchical Dirichlet Process (HDP) (Teh et al., 2006). However, those are with conventional clustering methods that apply on raw data. Recently, deep learning based clustering methods have drawn more and more attention. A simple two-stage approach is to first learn low-dimensional embeddings using unsupervised feature learning methods, and then perform cluster analysis on the embeddings. However, without any supervision, the representation learning do not necessarily reveal the true cluster structure of the data. DEC (Xie et al., 2016) is a method that simultaneously learns feature representations and cluster assignments through deep autoencoders. It gradually improves the clustering by driving the deep network to learn a better mapping. Improved Deep Embedded Clustering (Guo et al., 2017) improves DEC by keeping the decoder network and adding reconstruction loss to the original clustering loss in DEC. Variational deep embedding (Jiang et al., 2017) is a generative method that models the data generative process using a Gaussian mixture model combined with a VAE, and also performs joint learning of representations and clustering. Similarly, GMVAE (Dilokthanakul et al., 2016) performs joint learning of a GMM and a VAE, but instead generates the mixture components through neural networks. Deep clustering network (DCN) (Yang et al., 2017) is another one that jointly learns an autoencoder and performs k-means clustering. These joint learning methods consistently achieve better clustering results than conventional ones. The method proposed in (Yang et al., 2016) uses convolutional neural networks and jointly learns the representations and clustering in a recurrent framework. All these methods assume flat partitions over the data, and do not attempt the structure learning issue. An exception is hierarchical nonparametric variational autoencoders proposed in (Goyal et al., 2017). It uses nCRP as the prior for VAE to allow infinitely deep and branching tree hierarchy structure and focuses on learning hierarchy of concepts. However, it is still one partition over the data, only that the partitions in upper levels are more general partitions, while those in lower levels more fine-grained. Different from it, our work focuses on multifacets of clustering, for example, the model could make one partition based on identity of subjects, while another partition based on pose.
+
+# 3 THE PROPOSED METHOD
+
+In this section, we present the proposed latent tree variational autoencoder and the learning algorithms for joint representation learning and structure learning for multidimensional clustering.
+
+# 3.1 LATENT TREE VARIATIONAL AUTOENCODER
+
+Deep generative models assume that data $\mathbf { x }$ is generated from latent continuous variable $\mathbf { z }$ through some random process. The process consists of two steps: (1) a value $\mathbf { z }$ is generated from some prior distribution $p ( \mathbf { z } )$ ; (2) the observation $x$ is generated from the conditional distribution $p _ { \boldsymbol { \theta } } ( \mathbf { x } | \mathbf { z } )$ , which is parameterized through deep neural networks. Thus, it defines the joint distribution between
+
+![](images/6d0b434d991986a57615a54ec0bccda8c41c57d94632b929e8edcdd1f2ac02cb.jpg)  
+Figure 1: Latent Tree Variational Autoencoder
+
+![](images/5fda2f4f43615c251abdb445adf97d9754edcb8a075cd2d69b2172bb8bcf51f5.jpg)  
+Figure 2: Inference and gradient through message passing. Solid-arrows denote collecting message, and dashed-arrows denote distributing message.
+
+observation $\mathbf { x }$ and latent variable $\mathbf { z }$ :
+
+$$
+p ( \mathbf { x } , \mathbf { z } ) = p ( \mathbf { z } ) p _ { \theta } ( \mathbf { x } | \mathbf { z } )
+$$
+
+This process is hidden from our view, and we learn this process by maximizing the marginal loglikelihood $p ( \mathbf { x } )$ over the parameters $\theta$ and latent variable $\mathbf { z }$ from data. After the learning, the latent variable $\mathbf { z }$ can be regarded as the deep representations of $\mathbf { x }$ since it captures the most relevant information of $\mathbf { x }$ . Thus, the learning process is also called representation learning.
+
+In order to learn the latent structure of $\mathbf { z }$ , for example multidimensional cluster structure, we introduce a set of latent variables $Y _ { 1 } , . . . , Y _ { l }$ on top of $\mathbf { z }$ . A single $z$ or multiple $z$ ’s form a node $\mathbf { z } _ { b }$ . Suppose variables in $\mathbf { z }$ form $B$ nodes of ${ \bf z } _ { 1 } , \cdots , { \bf z } _ { B }$ . Each latent variable $Y$ may be only connected to a subset of nodes, and the dependency of each $\mathbf { z } _ { b }$ and its parent $Y$ is characterized by a conditional Gaussian distribution. Furthermore, the latent variables $Y _ { 1 } , . . . , Y _ { l }$ are connected to each other, and the dependency of a latent variable $Y$ on its parent $Y ^ { \prime }$ is characterized by a conditional distribution $P ( Y | Y ^ { \prime } )$ . This essentially forms a Bayesian network. And if we restrict the network to be treestructured, the $\mathbf { z }$ and $\mathbf { Y }$ together form a latent tree model (Zhang, 2004; Poon et al., 2010; 2013; Mourad et al., 2013; Pearl, 2014; Zhang & Poon, 2017) with $\mathbf { z }$ being the observed variables and $\mathbf { Y }$ being the latent variables. For multidimensional clustering, each latent variable $Y$ is taken to be a discrete variable, where each discrete state $y$ of $Y$ defines a cluster. Each latent variable $Y$ thus defines a facet partition over the data based on subset of attributes and multiple $Y$ ’s define multiple facets. Given a value $y$ of $Y$ , $\mathbf { z } _ { b }$ follows a conditional Gaussian distribution $\dot { P } ( \mathbf { z } _ { b } | y ) = \mathcal { N } ( \mu _ { y } , \bar { \Sigma } _ { y } )$ with mean vector $\mu _ { y }$ and covariance matrix $\Sigma _ { y }$ . Thus, each $\mathbf { z } _ { b }$ and its parent constitute a Gaussian mixture model (GMM). Suppose the parent of a node is denoted as $\pi ( \cdot )$ , the maginal distribution of $\mathbf { z }$ is defined as follows
+
+$$
+p ( \mathbf { z } ) = \sum _ { \mathbf { Y } } \prod _ { j = 1 } ^ { l } p ( y _ { j } | \pi ( Y _ { j } ) ) \prod _ { b = 1 } ^ { B } \mathcal { N } ( \mathbf { z } _ { b } | \mu _ { \pi ( \mathbf { z } _ { b } ) } , \boldsymbol { \Sigma } _ { \pi ( \mathbf { z } _ { b } ) } ) ,
+$$
+
+which sums over all possible combinations of $\mathbf { Y }$ states. As a matter of fact, a GMM is a Gaussian LTM that has only one latent variable connecting to all observed variables.
+
+Let the latent structure of $\mathbf { Y }$ be $s$ , defining the number of latent variables in $\mathbf { Y }$ , the number of discrete states in each variable $Y$ and the connectivity structure among all variables in $\mathbf { z }$ and $\mathbf { Y }$ . And let the parameters for all conditional probabilities in the latent structure be $\Theta$ . Both the latent structure $s$ and the latent parameters $\Theta$ are unknown. We aim to jointly learn data representations and the latent structure. The proposed LTVAE model is shown in Fig. 1. The latent structure $s$ are automatically learned from data and will be discussed in a later section.
+
+Due to the existence of the generation network, the inference of the model is intratable. Instead, we do amortized variational inference for the latent variable $\mathbf { z }$ by introducing an inference network (Kingma & Welling, 2014) and define an approximate posterior $q _ { \phi } ( { \bf z } | { \bf x } )$ . The evidence lower bound (ELBO) $\mathcal { L } _ { \mathrm { E L B O } }$ of the marginal loglikelihood of the data given $( \cal S , \Theta )$ is:
+
+$$
+\mathcal { L } _ { \mathrm { E L B O } } ( \mathbf { x } ) = \mathbb { E } _ { q _ { \phi } ( \mathbf { z } | \mathbf { x } ) } [ \log p _ { \theta } ( \mathbf { x } | \mathbf { z } ) ] + \mathbb { E } _ { q _ { \phi } ( \mathbf { z } | \mathbf { x } ) } [ \log \sum _ { \mathbf { y } } p _ { S } ( \mathbf { z } , \mathbf { y } ; \Theta ) ] + \mathbb { H } [ q _ { \phi } ( \mathbf { z } | \mathbf { x } ) ] ,
+$$
+
+where $\begin{array} { r } { \log \sum _ { \mathbf { y } } p _ { S } ( \mathbf { z } , \mathbf { y } ; \Theta ) } \end{array}$ is the marginal loglikelihood of the latent variable $\mathbf { z }$ under the latent tree model, and $\mathbb { H } [ \cdot ]$ is the entropy. The conditional generative distribution $p _ { \boldsymbol { \theta } } ( \mathbf { x } | \mathbf { z } )$ could be a Gaussian distribution if the input data is real-valued, or a Bernoulli distribution if binary, parameterized by the generation network. Using Monte Carlo sampling, the ELBO can be asymptotically estimated by
+
+$$
+\mathcal { L } _ { \mathrm { E L B O } } ( \mathbf { x } ) \simeq \frac { 1 } { M } \sum _ { i = 1 } ^ { M } \log p _ { \boldsymbol \theta } ( \mathbf { x } | \mathbf { z } ^ { ( i ) } ) + \log \sum _ { \mathbf { y } } p _ { \mathcal { S } } ( \mathbf { z } ^ { ( i ) } , \mathbf { y } ; \boldsymbol \Theta ) + \mathbb { H } [ q _ { \boldsymbol \phi } ( \mathbf { z } | \mathbf { x } ) ] ,
+$$
+
+where $\mathbf { z } ^ { ( i ) } \sim q _ { \phi } ( \mathbf { z } | \mathbf { x } )$ . The term $\mathbb { H } [ q _ { \phi } ( \mathbf { z } | \mathbf { x } ) ]$ can be computed analytically if we choose the form of $q _ { \phi } ( { \bf z } | { \bf x } )$ to be a Gaussian distribution $\begin{array} { r } { \mathcal { N } ( { \bf z } ; \mu _ { { \bf x } } , \sigma _ { { \bf x } } ) \colon \mathbb { H } [ q _ { \phi } ( { \bf z } | { \bf x } ) ] = \frac { J } { 2 } \log ( 2 \pi ) + \frac { 1 } { 2 } \sum _ { j = 1 } ^ { J } ( 1 + \log \sigma _ { j } ^ { 2 } ) . } \end{array}$ , where $J$ is the dimensionality of $\mathbf { z }$ .
+
+Furthermore, the marginal loglikelihood $\begin{array} { r } { \log \sum _ { \mathbf { y } } p _ { \mathcal { S } } ( \mathbf { z } ^ { ( i ) } , \mathbf { y } ; \Theta ) } \end{array}$ can be computed efficiently through message passing. Message passing is an efficient algorithm for inference in Bayesian networks (Koller & Friedman, 2009; Poon et al., 2013). In message passing, we first build a clique tree using the factors in the defined probability density. Because of the tree structure, each $\mathbf { z } _ { b }$ along with its parent form a clique with the potential $\psi ( \mathbf { z } _ { b } , y )$ being the corresponding conditional distribution. This is illustrated in Fig. 2. With the sampled $\mathbf { z } ^ { ( i ) }$ , we can compute the message $\psi ^ { \prime } ( y )$ by absorbing the evidence from $\mathbf { z }$ . During collecting message phase, the message $\psi ^ { \prime } ( y )$ are sent towards the pivot. After receiving all messages, the pivot distributes back messages towards all $\mathbf { z }$ . Both the posterior of $\mathbf { Y }$ and the marginal loglikelihood of $\mathbf { z } ^ { ( i ) }$ thus can be computed in the final normalization step.
+
+3.2 PARAMETER LEARNING THROUGH GRADIENT DESCENT AND STEPWISE EM WITH MESSAGE PASSING
+
+In this section, we propose efficient learning algorithms for LTVAE through gradient descent and stepwise EM with message passing.
+
+Given the latent tree model $( \cal S , \Theta )$ , the parameters of neural networks can be efficiently optimized through stochastic gradient descent (SGD). However, in order to learn the model, it is important to efficiently compute the gradient of the marginal loglikelihood $\log p _ { S } ( \mathbf { z } ; \Theta )$ from the latent tree model, the third term in Eq. 4. Here, we propose an efficient method to compute gradient through message passing. Let $\mathbf { z } _ { b }$ be the variables that we want to compute gradient with respect to, and let $Y _ { b }$ be the parent node. The marginal loglikelihood of full $\mathbf { z }$ can be written as
+
+$$
+\log p _ { \mathcal { S } } ( \mathbf { z } ; \Theta ) = \log [ \sum _ { y _ { b } } \mathcal { N } ( \mathbf { z } _ { b } | \mu _ { y _ { b } } , \boldsymbol { \Sigma } _ { y _ { b } } ) f ( y _ { b } ) ] ,
+$$
+
+where $f ( y _ { b } )$ is the collection of all the rest of the terms not containing $\mathbf { z } _ { b }$ . The gradient $\mathbf { g } _ { \mathbf { z } _ { b } }$ of the marginal loglikelihood $\log p s ( \mathbf { z } ; \Theta )$ w.r.t $\mathbf { z } _ { b }$ thus can be computed as
+
+$$
+\mathbf { g } _ { \mathbf { z } _ { b } } = \frac { 1 } { p _ { S } ( \mathbf { z } ; \boldsymbol { \Theta } ) } \frac { \partial \sum _ { y _ { b } } f ( y _ { b } ) \mathcal { N } ( \mathbf { z } _ { b } | \mu _ { y _ { b } } , \boldsymbol { \Sigma } _ { y _ { b } } ) } { \partial \mathbf { z } _ { b } } = \sum _ { y _ { b } } p ( y _ { b } | \mathbf { z } ) \frac { \partial \log [ f ( y _ { b } ) \mathcal { N } ( \mathbf { z } _ { b } | \mu _ { y _ { b } } , \boldsymbol { \Sigma } _ { y _ { b } } ) ] } { \partial \mathbf { z } _ { b } }
+$$
+
+where $p ( \boldsymbol { y } _ { b } | \mathbf { z } )$ is the posterior probability of $y _ { b }$ and can be computed efficiently with message passing as described in the previous section. The detailed derivation is in Appendix E. Since $\mathbf { z } = [ \bar { \mathbf { z } _ { 1 } } , \bar { \mathbf { . . . , z } _ { B } } ]$ , we have
+
+$$
+\frac { \partial \log p ( \mathbf { z } ) } { \partial \mathbf { z } } = \left[ \frac { \partial \log p ( \mathbf { z } ) } { \partial \mathbf { z } _ { 1 } } , . . . , \frac { \partial \log p ( \mathbf { z } ) } { \partial \mathbf { z } _ { B } } \right] = \left[ \mathbf { g } _ { \mathbf { z } _ { 1 } } , . . . , \mathbf { g } _ { \mathbf { z } _ { B } } \right] .
+$$
+
+With the efficient computation of the third term in Eq. 4 and its gradient w.r.t $\mathbf { z }$ through message passing, the parameters of inference network and generation network can be efficiently optimized through SGD.
+
+In order to jointly learn the parameters of the latent tree $\Theta$ , we propose Stepwise EM algorithm based on mini-batch of data. Specifically, we maximize the third term in Eq. 4, i.e. the marginal loglikelihood of $\mathbf { z }$ under the latent tree. In the Stepwise E-step, we compute the distributions $\bar { P ( } y , y ^ { \prime } | \mathbf { z } , \theta ^ { ( t - 1 ) } )$ and $P ( y | \mathbf { z } , \theta ^ { ( t - 1 ) } )$ for each latent node $Y$ and its parent $Y ^ { \prime }$ . In the Stepwise
+
+![](images/5242bd844d379aeb381ed06e3d72093a6b1e707f80ef08bb209cd29202145d77.jpg)  
+Figure 3: Structure search operators. The digits above the nodes denote the number of discrete states. Node deletion, state deletion and unpouching are the inverse of node insertion, state insertion and pouching, respectively.
+
+M-step, we estimate the new parameter $\theta ^ { ( t ) }$ . Let $\mathbf { s } ( \mathbf { z } , \mathbf { y } )$ be a vector the sufficient statistics for a single data case. Let $\bar { \mathbf { s } } = \mathbb { E } _ { p s ( \mathbf { y } | \mathbf { z } ; \boldsymbol { \Theta } ) } [ \mathbf { s } ( \mathbf { z } , \mathbf { y } ) ]$ be the expected sufficient statistics for the data case, where the expectation is w.r.t the posterior distribution of $\mathbf { y }$ with current parameter. And let $\begin{array} { r } { \boldsymbol { \mu } = \sum _ { i = 1 } ^ { N } \bar { \mathbf { s } } _ { i } } \end{array}$ be the sum of the expected sufficient statistics. The update of the parameter $\Theta$ is performed as follows:
+
+$$
+\begin{array} { r l } & { ~ \bar { \mathbf { s } } _ { i } ^ { t } = \mathbb { E } _ { p s ( \mathbf { y } _ { i } \mid \mathbf { z } _ { i } ; \boldsymbol { \Theta } ^ { t } ) } [ \mathbf { s } ( \mathbf { z } _ { i } , \mathbf { y } _ { i } ) ] } \\ & { ~ \mu ^ { t + 1 } = \mu ^ { t } + \eta ( \bar { \mathbf { s } } _ { i } ^ { t } - \mu ^ { t } ) } \\ & { \boldsymbol { \Theta } ^ { t + 1 } = \underset { \boldsymbol { \Theta } } { \arg \operatorname* { m a x } } l ( \mu ^ { t + 1 } , \boldsymbol { \Theta } ) , } \end{array}
+$$
+
+where $\eta$ is the learning rate and $l$ is the complete data loglikelihood. Each iteration of update of LTVAE thus is composed of one iteration of gradient descent update for the neural network parameters and one iteration of Stepwise EM update for the latent tree model parameters with a mini-batch of data.
+
+# 3.3 STRUCTURE LEARNING
+
+For the latent structure $s$ , there are four aspects need to determine: the number of latent variables, the cardinalities of latent variables, the connectivities among variables. We aim at finding the model $m ^ { * }$ that maximizes the BIC score (Schwarz et al., 1978; Koller & Friedman, 2009):
+
+$$
+B I C ( m | \mathcal { D } ) = \log P ( \mathcal { D } | m , \theta ^ { * } ) - \frac { d ( m ) } { 2 } \log N ,
+$$
+
+where $\theta ^ { * }$ is the MLE of the parameters and $d ( m )$ is the number of independent parameters. The first term is known as the likelihood term. It favors models that fit data well. The second term is known as the penalty term. It discourages complex models. Hence, the BIC score provides a tradeoff between model fit and model complexity. To this end, we perform systematic searching to find a structure with a high BIC score. We use the hill-climing algorithm to search for $m ^ { * }$ as in (Poon et al., 2010; 2013), and define 7 search operators: node introduction (NI) and node deletion (ND) to introduce new latent nodes and delete existing nodes, state introduction (SI) and state deletion (SD) to add a new state and delete a state for existing nodes, node relocation (NR) to change links of existing nodes, pouching (PO) and unpouching (UP) operators to combine nodes into a single node and separate variables from a node.. The structure search operators are shown in Fig. 3. Each operator produces a set of candidates from existing structure, and the best candidate is picked if it improves the previous one. To reduce the number of possible search candidates, we first perform SI, NI and PO to expand the structure and pick the best model. Then we perform NR to adjust the best model. Finally, we perform UP, ND and SD to simplify the current best structure and pick the best one. Acceleration techniques (Poon et al., 2013) are adopted that make the algorithm efficient enough. The structure learning is performed iteratively together with the parameter learning of neural networks.
+
+The overall learning algorithm is illustrated in Algorithm 1. Starting from a pretrained model, we iteratively improve the structure and parameters of latent tree model while learning the representations of data through neural network in a greedy manner. Using current structure $S ^ { t }$ as the initial structure,
+
+# Algorithm 1 Learning Latent Tree Variational Autoencoder
+
+<table><tr><td>Input:data D,z dim,neural networks,E 0,,S°,Θ° ← pretrain(D) repeat fore=1toEdo</td></tr><tr><td>for each minibatchX in D do Compute q(z|μx,Ox)</td></tr><tr><td>Sample z(i) ~q(z|μx,x) Compute log ps(z; ) and 0log ps(z) from Eq.5 and 7 dz</td></tr><tr><td>Compute ELBO from Eq. 4 0,Φ ← Back-propagation and SGD step</td></tr><tr><td>← StepwiseEM(z()) end for</td></tr><tr><td>end for Dz↑μD</td></tr><tr><td></td></tr><tr><td></td></tr><tr><td>repeat</td></tr><tr><td>S*,Θ* ← SearchWith(St-1,Θt-1,{SI,NI,PO})</td></tr><tr><td>S*,Θ* ← SearchWith(S*,Θ* {NR}) St,Θt ← SearchWith(S*,Θ*,{UP,ND,SD}) until BIC(St,0t|Dz)≤BIC(St-1,0t-1|Dz)</td></tr></table>
+
+we search for a better model. With new latent tree model, we optimize for a better representation until convergence.
+
+# 4 EXPERIMENTS
+
+# 4.1 SYNTHETIC-DATA DEMONSTRATION
+
+We first demonstrate the effectiveness of the proposed method through synthetic data. Assume that the data points have two facets $Y _ { 1 }$ and $Y _ { 2 }$ , where each facet controlls a subset of attributes (e.g. two-dimensional domain) and defines one partition over the data. This four-dimensional domain $\mathbf { z } = \{ z _ { 1 } , z _ { 2 } , z _ { 3 } , z _ { 4 } \}$ is a latent representation which we do not observe. What we observe is $\mathbf { x } \in \mathbb { R } ^ { 1 0 0 }$ that is obtained via the following non-linear transformation:
+
+$$
+\mathbf { x } = \sigma ( U \sigma ( W \mathbf { z } ) ) ,
+$$
+
+where $W \in \mathbb { R } ^ { 1 0 \times 4 }$ and $U \in \mathbb { R } ^ { 1 0 0 \times 1 0 }$ are matrices whose entries follow the zero-mean unit-variance i.i.d. Gaussian distribution, $\sigma ( \cdot )$ is a sigmoid function to introduce nonlinearity. The generative model is shown in Fig. 4 (a). We define two clusters in facet $Y _ { 1 }$ and two clusters in facet $Y _ { 2 }$ , and generate 5,000 samples of $\mathbf { x }$ . Under the above generative model, recovering the two facets $Y _ { 1 }$ and $Y _ { 2 }$ structure and the latent $z$ domain from the observation of $\mathbf { x }$ seems very challenging. All previous DNN-based methods $\mathbf { \Gamma } ( \mathbf { A E + G M M }$ , DEC, DCN, etc.) are only able to discover one-facet of clustering (i.e. one partition over the data), and none of these is applicable to solve such a multidimensional clustering problem. Fig. 4 (b) shows the results of the proposed method. As one can see, the LTVAE successfully discovers the true superstructure of $Y _ { 1 }$ and $Y _ { 2 }$ . The 2-d plot of $z _ { 1 }$ and $z _ { 2 }$ shows the separable latent space clusters under facet $Y _ { 1 }$ , and it matches the ground-truth cluster assignments. Additionally, the 2-d plot of $z _ { 3 }$ and $z _ { 4 }$ shows another separable clusters under facet $Y _ { 2 }$ , and it also matches the ground-truth cluster assignments well in the other facet.
+
+# 4.2 REAL-DATA EXPERIMENT SETUP
+
+We evaluate the proposed LTVAE model on two image datasets and two other datasets, and compare it against other deep learning based clustering algorithms, including two-stage methods, $\mathbf { A E { + } G M M }$ and $\mathbf { V A E + G M M }$ , which first learn AE/VAE (Kingma & Welling, 2014) models then construct a GMM on top of them, and joint learning methods, DEC (Xie et al., 2016) and DCN (Yang et al., 2017). The datasets include MNIST, STL-10, Reuters (Xie et al., 2016; Jiang et al., 2017) and the Heterogeneity Human Activity Recognition (HHAR) dataset. When evaluating the clustering performance, for fair of comparison, we follow previous works (Xie et al., 2016; Yang et al., 2017) and use the network structures of $d - 5 0 0 - 5 0 0 - 2 0 0 0 - 1 0$ for the encoder network and $1 0 - 2 0 0 0 -$ $5 0 0 - 5 0 0 - d$ for the decoder network for all datasets, where $d$ is the data-space dimension, which varies among datasets. All layers are fully-connected. We follow the pretraining procedure as in (Xie et al., 2016). We first perform greedy layer-wise pretraining in denoising autoencoder manner, then stack all layers to form deep autoencoder. The deep autoencoder is further finetuned to minimize the reconstruction loss. The weights of the deep autoencoder are used to intialize the weights of encoder and decoder networks of above methods. After the pretraining, we optimze the objectives of those methods. For DEC and DCN, we use the same hyperparameter settings as the original papers. When initializing the cluster centroids for DEC and DCN, we perform 10 random restarts and pick the results with the best objective value for $k$ -means/GMM. For the proposed LTVAE, we use Adam optimzer (Kingma & Ba, 2015) with initial learning rate of 0.001 and mini-batch size of 128. For Stepwise EM, we set the learning rate to be 0.01. As in Algorithm 1, we set $E = 5$ , i.e. we update the latent tree model every 5 epochs. When optimizing the candidate models during structure search, we perform 10 random restarts and train with EM for 200 iterations.
+
+![](images/e672589b914b2a28cffd51a0a4a0d6081efff6a689a04a75ded1c725ef434333.jpg)  
+Figure 4: (a) The generative process of synthetic data; (b) The discovered multidimensional superstructure and the latent space (different colors denote different ground truth clusters in each facet.)
+
+Table 1: Test data loglikelihood for various datasets.   
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>MNIST</td><td rowspan=1 colspan=1>STL</td><td rowspan=1 colspan=1>Reuters</td><td rowspan=1 colspan=1>HHAR</td></tr><tr><td rowspan=1 colspan=1>VAE</td><td rowspan=1 colspan=1>-86.64±0.20</td><td rowspan=1 colspan=1>-743.72±0.51</td><td rowspan=1 colspan=1>-1312.40±1.24</td><td rowspan=1 colspan=1>-17.88±0.26</td></tr><tr><td rowspan=1 colspan=1>IWAE</td><td rowspan=1 colspan=1>-85.39±0.13</td><td rowspan=1 colspan=1>-742.43±1.10</td><td rowspan=1 colspan=1>-1254.29±6.95</td><td rowspan=1 colspan=1>-16.53±0.16</td></tr><tr><td rowspan=1 colspan=1>LTVAE</td><td rowspan=1 colspan=1>-84.75±0.14</td><td rowspan=1 colspan=1>-619.04±7.88</td><td rowspan=1 colspan=1>-1245.71±4.45</td><td rowspan=1 colspan=1>-13.65±0.53</td></tr></table>
+
+# 4.3 TEST LOGLIKELIHOOD
+
+We first show that, by using the marginal loglikelihood defined by the latent tree model as the prior, LTVAE better fits the data than conventional VAE and importance weighted autoencoders (IWAE) (Burda et al., 2016). While alternative quantitative criteria have been proposed (Bounliphone et al., 2016; Im et al., 2016; Salimans et al., 2016) for generative models, log-likelihood of held-out test data remains one of the most important measures of a generative model’s performance (Kingma & Welling, 2014; Burda et al., 2016; Wu et al., 2017; Goyal et al., 2017). For comparison, we approximate true loglikelihood $\mathcal { L } _ { 5 0 0 0 }$ using importance sampling (Burda et al., 2016): ${ \mathcal { L } } _ { k } ( \mathbf { x } ) =$ $\begin{array} { r } { \log { \frac { 1 } { k } } \sum _ { i = 1 } ^ { k } \frac { p _ { \theta } ( \mathbf { x } , \mathbf { z } ^ { ( i ) } ) } { q _ { \phi } ( \mathbf { z } ^ { ( i ) } | \mathbf { x } ) } } \end{array}$ , where $\mathbf { z } ^ { ( i ) } \sim q _ { \phi } ( \mathbf { z } | \mathbf { x } )$ . The results for all datasets are shown in Table 1. The proposed LTVAE obtains a higher test data loglikelihood and ELBO, implying that it can better model the underlying complex data distribution embedded in the image data.
+
+![](images/1d233589357b0631e59d50b485f75fabf632ff9d82873f460dacc04ee477949a.jpg)  
+Figure 5: Two facet clustering results from LTVAE are in (a) digit identity and (b) shape and pose. Each row contains the top 10 scoring elements from one cluster. (c) shows the pose variations by fixing the digit cluster in facet 1 and changing the cluster in facet 2. It can be seen that up-right, left-tilted and right-tilted images of the same digits are clearly recognizable.
+
+Table 2: Clustering Accuracy of clustering results.   
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>MNIST</td><td rowspan=1 colspan=1>STL-10</td><td rowspan=1 colspan=1>Reuters</td><td rowspan=1 colspan=1>HHAR</td></tr><tr><td rowspan=1 colspan=1>AE+GMM</td><td rowspan=1 colspan=1>82.18%</td><td rowspan=1 colspan=1>79.83%</td><td rowspan=1 colspan=1>68.68%</td><td rowspan=1 colspan=1>78.90%</td></tr><tr><td rowspan=1 colspan=1>VAE+GMM</td><td rowspan=1 colspan=1>76.87%</td><td rowspan=1 colspan=1>79.49%</td><td rowspan=1 colspan=1>65.85%</td><td rowspan=1 colspan=1>67.91%</td></tr><tr><td rowspan=1 colspan=1>DEC</td><td rowspan=1 colspan=1>84.30%</td><td rowspan=1 colspan=1>80.62%</td><td rowspan=1 colspan=1>74.32%</td><td rowspan=1 colspan=1>79.86%</td></tr><tr><td rowspan=1 colspan=1>DCN</td><td rowspan=1 colspan=1>83.32%</td><td rowspan=1 colspan=1>85.88%</td><td rowspan=1 colspan=1>75.05%</td><td rowspan=1 colspan=1>81.26%</td></tr><tr><td rowspan=1 colspan=1>LTVAE</td><td rowspan=1 colspan=1>86.32%</td><td rowspan=1 colspan=1>90.00%</td><td rowspan=1 colspan=1>80.96%</td><td rowspan=1 colspan=1>85.00%</td></tr></table>
+
+# 4.4 MULTIFACET CLUSTERING
+
+The most important features of the proposed model are that it can perform variable selection for model-based clustering, leading to multiple facets clustering.
+
+We use the standard unsupervised evaluation metric and protocols for evaluations and comparisons to other algorithms (Yang et al., 2010). For baseline algorithms we set the number of clusters to the number of ground-truth categories. While for LTVAE, it automatically determines the number of facets and latent superstructure through structure learning. We evaluate performance with unsupervised clustering accuracy $( A C C )$ :
+
+$$
+A C C = \operatorname* { m a x } _ { m } \frac { \sum _ { i = 1 } ^ { n } \mathbf { 1 } \left\{ l _ { i } = m ( c _ { i } ) \right\} } { n } ,
+$$
+
+where $l _ { i }$ is the groundtruth label, $c _ { i }$ is the cluster assignment produced by the algorithm, and $m$ ranges over all possible mappings between clusters and labels. Table 2 show the quantitative clustering results compared with previous works. With $\mathbf { z }$ dimension of small value like 10, LTVAE usually discovers only one facet. It can be seen the, for MNIST dataset LTVAE achieves clustering accuracy of $8 6 . 3 2 \%$ , better than the results of other methods. This is also the case for STL-10, Reuters and HHAR.
+
+More importantly, the proposed LTVAE does not just give one partition over the data. Instead, it explains the data in multi-faceted ways. Unlike previous clustering experiments, for this experiment, we choose the z dimension to be 20. Fig. 5 shows the two facet clustering results for MNIST. It can be seen that facet 1 gives quite clean clustering over the identity of the digits and the ten digits are well separated. On the other hand, facet 2 gives a more grand partition based on the shape and pose. Note how up-right $" 4 > "$ and “9” are similar, and how tilted “4”,“7” and “9” are similar. The facet meanings are more evident in Fig. 5 (c). Fig. 6 shows four facets discovered for the STL-10 dataset. Although it is hard to characterize precisely how the facets differ from each other, there are visible patterns. For example, the cats, monkeys and birds in facet 2 have clearly visible eyes, while this is not always true in facet 1. The deers in facet 2 are all showing their antlers/ears, while this is not true in facet 3. In facet 2 we see frontal views of cars, while in facets 1 and 3 we see side view of cars. In facet 1, each cluster consists of the same types of objects/animal. In facet 3/4, images in the same cluster do not necessarily show the same type of objects/animals. However, they have similar overall feel.
+
+![](images/ff99c4bc8ecd2f35cfc249876f70070a5f186e93efe69e4180f070a4bf2f59d6.jpg)  
+Figure 6: Clustering results from LTVAE for STL-10 dataset. Each row contains the top 5 scoring elements from one cluster.
+
+![](images/c09285f05ed4a5f8198b024ddb48f70e4a9e48fdbbb909c8d2925f06a53057f9.jpg)  
+Figure 7: The digits generated by the proposed model. Digits in the same row come from the same latent code of the latent tree.
+
+# 4.5 IMAGE GENERATION
+
+Since the structure of the data in latent space is automatically learned through the latent tree, we can sample the data in a more structured way. One way is through ancestral sampling, where we first sample the root of the latent tree and then hierarchically sample the children variables to get $\mathbf { z }$ , from which the images can be generated through generation network. The other way is to pick one component from the Gaussian mixture and sample $\mathbf { z }$ from that component. This produces samples from a particular cluster. Fig. 7 shows the samples generated in this way. As it can be seen, digits sampled from each component has clear semantic meaning and belong to the same category. Whereas, the samples generated by VAE does not have such structure. Conditional image generation can also be performed to alter the attributes of the same digit as shown in Appendix B.
+
+# 5 DISCUSSIONS
+
+LTVAE learns the dependencies among latent variables $\mathbf { Y }$ . In general, latent variables are often correlated. For example, the social skills and academic skills of a student are generally correlated. Therefore, its better to model this relationship to better fit the data. Experiments show that removing such dependencies in LTVAE models results in inferior data loglikelihood.
+
+In this paper, for the inference network, we simply use mean-field inference network with same structure as the generative network (Kingma & Welling, 2014). However, the limited expressiveness of the mean-field inference network could restrict the learning in the generative network and the quality of the learned model (Webb et al., 2018; Rainforth et al., 2018; Cremer et al., 2018). Using a faithful inference network structure as in (Webb et al., 2018) to incorporate the dependencies among latent variables in the posterior, for example one parameterized with masked autoencoder distribution estimator (MADE) model (Germain et al., 2015), could have a significant improvement in learning. We leave it for future investigation.
+
+# 6 CONCLUSIONS
+
+In this paper, we propose an unsupervised learning method, latent tree variational autoencoder (LTVAE), which simultaneously performs representation learning and multidimensional clustering. Different from previous deep learning based clustering methods, LTVAE learns latent embeddings from data and discovers multi-facet clustering structure based on subsets of latent features rather than one partition over data. Experiments show that the proposed method achieves state-of-the-art clustering performance and reals reasonable multifacet structures of the data.
+
+# ACKNOWLEDGMENTS
+
+Research on this article was supported by Hong Kong Research Grants Council under grants 16212516 and 16202118.
+
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+Jianwei Yang, Devi Parikh, and Dhruv Batra. Joint unsupervised learning of deep representations and image clusters. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 5147–5156, 2016.
+
+Yi Yang, Dong Xu, Feiping Nie, Shuicheng Yan, and Yueting Zhuang. Image clustering using local discriminant models and global integration. IEEE Transactions on Image Processing, 19(10): 2761–2773, 2010.
+
+Nevin L Zhang. Hierarchical latent class models for cluster analysis. Journal of Machine Learning Research, 5(6):697–723, 2004.
+
+Nevin L Zhang and Leonard KM Poon. Latent tree analysis. In AAAI, pp. 4891–4898, 2017.
+
+# A SUPERSTRUCTURES
+
+For the MNIST dataset, the conditional probability between identity facet $Y _ { 1 }$ ( $\mathbf { \bar { X } } ^ { \prime }$ -axis) and pose facet $Y _ { 2 }$ (y-axis) is shown in Fig. 8. It can be seen that a cluster in $Y _ { 1 }$ facet could correspond to multiple clusters in $Y _ { 2 }$ facet due to the conditional probability, e.g. cluster 0, 4, 5, 11 and 12. However, not all clusters in $Y _ { 2 }$ facet are possible for a given cluster in $Y _ { 1 }$ facet.
+
+![](images/d997c4363a0c39b11d777cc78d967c979fc2920e26ce9822474c2bcb0761d59c.jpg)  
+Figure 8: Conditional probability of $Y _ { 1 }$ and $Y _ { 2 }$ for the two facets of MNIST discovered by LTVAE.
+
+# B CONDITIONAL IMAGE GENERATION
+
+Here we show more results on conditional image generation. Interestingly, with LTVAE, we can change the original images by fixing variables in some facet and sampling in other facets. For example, in MNIST we can fix the variables in identity facet and change the pose of the digit by sampling in the pose facet. Fig. 9 shows the samples generated in this way. As it can be seen, the pose of the input digits are changed in the samples generated by the proposed method.
+
+![](images/305531c7e4bf0fba99c111714fa1b632f5f0caa852d4e3492a11f51cc6df7b31.jpg)  
+Figure 9: Image generation. Left are the original image. Right are generated with the proposed model by fixing the variables in identity facet and sampling the variables in the pose facet. Digits in the same row come from the same latent code of the latent tree.
+
+# C COMPUTATIONAL TIME
+
+We compare the computational time of the proposed LTVAE w/ structure learning and that w/ fixed structure. For LTVAE with fixed structure, we fixed the structure of the latent tree model to be a single $Y$ connecting to all $z \mathbf { S }$ , in which each $z$ node consists of single $z$ variable.
+
+Table 3: Computational time (s) w/ and w/o structure learning.   
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>MNIST</td><td rowspan=1 colspan=1>STL-10</td><td rowspan=1 colspan=1>Reuters</td><td rowspan=1 colspan=1>HHAR</td></tr><tr><td rowspan=1 colspan=1>LTVAE w/ structure learning</td><td rowspan=1 colspan=1>7,592</td><td rowspan=1 colspan=1>3,251</td><td rowspan=1 colspan=1>5,756</td><td rowspan=1 colspan=1>5,693</td></tr><tr><td rowspan=1 colspan=1>LTVAE w/ fixed structure</td><td rowspan=1 colspan=1>3,197</td><td rowspan=1 colspan=1>542</td><td rowspan=1 colspan=1>1,442</td><td rowspan=1 colspan=1>1,021</td></tr></table>
+
+# D SIMILARITY BETWEEN LEARNED FACETS
+
+Different facets learned by LTVAE might have some overlaps among each other. Here we make quantitative comparison among different facets based on the cluster assignments in each facet. We evaluate the similarity between two clusterings $Y _ { 1 }$ and $Y _ { 2 }$ using normalized mutual information $N M I ( Y _ { 1 } ; Y _ { 2 } )$ . The NMI is given by
+
+$$
+N M I ( Y _ { 1 } ; Y _ { 2 } ) = \frac { I ( Y _ { 1 } ; Y _ { 2 } ) } { \sqrt { H ( Y _ { 1 } ) H ( Y _ { 2 } ) } } ,
+$$
+
+where $I ( Y _ { 1 } ; Y _ { 2 } )$ is the mutual information between $Y _ { 1 }$ and $Y _ { 2 }$ and $H ( V )$ is the entropy of a variable $V$ . These quantities can be computed from $P ( Y _ { 1 } ; Y _ { 2 } )$ , which in turn is estimated by $P ( Y _ { 1 } ; Y _ { 2 } ) =$ $\begin{array} { r l } { { } } & { { } { \frac { 1 } { N } } \sum _ { i = 1 } ^ { N } P ( Y _ { 1 } | \mathbf { d _ { i } } ) P ( Y _ { 2 } | \mathbf { d _ { i } } ) } \end{array}$ , where $\mathbf { d } _ { 1 } , \cdots , \mathbf { d } _ { N }$ are the samples in the test data.
+
+Table 4: NMI between different facets learned by LTVAE and groundtruth for MNIST dataset.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Groundtruth</td><td rowspan=1 colspan=1>Facet 1</td><td rowspan=1 colspan=1>Facet 2</td></tr><tr><td rowspan=1 colspan=1>Groundtruth</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0.825</td><td rowspan=1 colspan=1>0.574</td></tr><tr><td rowspan=1 colspan=1>Facet 1</td><td rowspan=1 colspan=1>0.825</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0.682</td></tr><tr><td rowspan=1 colspan=1>Facet 2</td><td rowspan=1 colspan=1>0.574</td><td rowspan=1 colspan=1>0.682</td><td rowspan=1 colspan=1>1</td></tr></table>
+
+Table 5: NMI between different facets learned by LTVAE and groundtruth for STL dataset.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Groundtruth</td><td rowspan=1 colspan=1>Facet1</td><td rowspan=1 colspan=1>Facet 2</td><td rowspan=1 colspan=1>Facet3</td><td rowspan=1 colspan=1>Facet 4</td></tr><tr><td rowspan=1 colspan=1>Groundtruth</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0.8613</td><td rowspan=1 colspan=1>0.6279</td><td rowspan=1 colspan=1>0.5758</td><td rowspan=1 colspan=1>0.5536</td></tr><tr><td rowspan=1 colspan=1>Facet 1</td><td rowspan=1 colspan=1>0.8613</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0.6962</td><td rowspan=1 colspan=1>0.6314</td><td rowspan=1 colspan=1>0.6251</td></tr><tr><td rowspan=1 colspan=1>Facet 2</td><td rowspan=1 colspan=1>0.6279</td><td rowspan=1 colspan=1>0.6962</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0.4675</td><td rowspan=1 colspan=1>0.5031</td></tr><tr><td rowspan=1 colspan=1>Facet 3</td><td rowspan=1 colspan=1>0.5758</td><td rowspan=1 colspan=1>0.6314</td><td rowspan=1 colspan=1>0.4675</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0.5885</td></tr><tr><td rowspan=1 colspan=1>Facet 4</td><td rowspan=1 colspan=1>0.5536</td><td rowspan=1 colspan=1>0.6251</td><td rowspan=1 colspan=1>0.5031</td><td rowspan=1 colspan=1>0.5885</td><td rowspan=1 colspan=1>1</td></tr></table>
+
+# E DERIVATION OF GRADIENT
+
+Here we give detailed derivation of Equation 6. The gradient $\mathbf { g } _ { \mathbf { z } _ { b } }$ of the marginal loglikelihood $\log p s ( { \bf z } ; \Theta )$ w.r.t $\mathbf { z } _ { b }$ thus can be computed as
+
+$$
+\begin{array} { r l } { \mathcal { Q } _ { \kappa , \kappa } = } & { \frac { \partial \langle \Phi _ { \kappa } | \phi _ { \sigma } \rangle \langle \kappa \Theta \rangle } { \partial \mathcal { L } _ { \kappa } } } \\ & { = \frac { \partial } { \partial \kappa _ { \kappa } \langle \kappa \Theta \rangle } _ { \kappa } \frac { \partial \langle \Phi _ { \kappa } | \phi _ { \sigma } \rangle _ { \kappa } } { \partial \mathcal { L } _ { \kappa } } } \\ & { = \frac { 1 } { \rho _ { \kappa } \langle \kappa \Theta \rangle } _ { \kappa } \frac { \partial \sum _ { \kappa } f \langle \Phi _ { \kappa } \rangle \langle \kappa \Theta _ { \kappa } | \phi _ { \kappa } \rangle _ { \kappa } \langle \kappa \Theta _ { \kappa } \rangle } { \partial \mathcal { L } _ { \kappa } } } \\ & { = \sum _ { \kappa } \frac { \partial } { \rho _ { \kappa } \langle \kappa \Theta \rangle } _ { \kappa } \frac { \partial | f \langle \Phi _ { \kappa } \rangle \langle \kappa | \Theta _ { \kappa } \rangle _ { \kappa } \langle \kappa \Theta _ { \kappa } \rangle | } { \partial \mathcal { L } _ { \kappa } } } \\ & { = \sum _ { \kappa } \frac { f \langle \Phi _ { \kappa } | \Phi \rangle \langle \kappa | \Phi _ { \kappa } | \rho _ { \kappa } \rangle _ { \kappa } \langle \kappa \Theta _ { \kappa } | \phi _ { \kappa } \rangle _ { \kappa } } { \partial \mathcal { L } _ { \kappa } \langle \kappa \Theta \rangle } \frac { \partial | f \langle \Phi _ { \kappa } | \mathcal { N } \langle \kappa | \Phi _ { \kappa } \rangle _ { \kappa } \langle \kappa \Theta _ { \kappa } \rangle _ { \kappa } | } { \partial \mathcal { L } _ { \kappa } } } \\ & { = \sum _ { \kappa } \frac { f \langle \Phi _ { \kappa } | \Phi \rangle \langle \kappa | \Phi _ { \kappa } | \rho _ { \kappa } \rangle _ { \kappa } \langle \kappa | \Phi _ { \kappa } \rangle _ { \kappa } \langle \kappa | \Phi _ { \kappa } \rangle _ { \kappa } } { \partial \mathcal { L } _ { \kappa } } } \\ & { = \sum _ { \kappa } \rho _ { \kappa } | z | \rho _ { \kappa } \langle | \Phi _ { \kappa } | \rho _ { \kappa } \rangle _ { \kappa } \langle | \Phi _ { \kappa } | \rho _ { \kappa } \rangle _ { \kappa } } \\ &  = \sum _ { \kappa } ^ { \kappa } \rho _ { \kappa } | z | \rho _ { \kappa } \langle \kappa | \Phi _ { \kappa } \rangle _ { \kappa }  \end{array}
+$$
+
+where $p ( \boldsymbol { y } _ { b } | \mathbf { z } )$ is the posterior probability of $y _ { b }$ and can be computed efficiently with message passing as described in the previous section. Note that
+
+$$
+p ( y _ { b } | \mathbf { z } ) = \frac { f ( y _ { b } ) \mathcal { N } ( \mathbf { z } _ { b } | \mu _ { y _ { b } } , \Sigma _ { y _ { b } } ) } { p s ( \mathbf { z } ; \boldsymbol { \Theta } ) }
+$$
+
+is valid due to $\begin{array} { r } { p _ { S } ( \mathbf { z } ; \Theta ) = \sum _ { y _ { b } } \mathcal { N } ( \mathbf { z } _ { b } | \mu _ { y _ { b } } , \boldsymbol { \Sigma } _ { y _ { b } } ) f ( y _ { b } ) } \end{array}$ and the Bayes rule.

parse/train/SJgNwi09Km/SJgNwi09Km_content_list.json

@@ -0,0 +1,1723 @@
+[
+    {
+        "type": "text",
+        "text": "LEARNING LATENT SUPERSTRUCTURES IN VARIATIONAL AUTOENCODERS FOR DEEP MULTIDIMENSIONAL CLUSTERING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
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+            171
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Xiaopeng $\\mathbf { L i } ^ { 1 }$ , Zhourong Chen1, Leonard K. M. Poon2 and Nevin L. Zhang1 ",
+        "bbox": [
+            184,
+            194,
+            710,
+            210
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Department of Computer Science and Engineering The Hong Kong University of Science and Technology 2 Department of Mathematics & Information Technology The Education University of Hong Kong {xlibo,zchenbb,lzhang}@cse.ust.hk, kmpoon@eduhk.hk ",
+        "bbox": [
+            184,
+            212,
+            671,
+            280
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            316,
+            544,
+            332
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We investigate a variant of variational autoencoders where there is a superstructure of discrete latent variables on top of the latent features. In general, our superstructure is a tree structure of multiple super latent variables and it is automatically learned from data. When there is only one latent variable in the superstructure, our model reduces to one that assumes the latent features to be generated from a Gaussian mixture model. We call our model the latent tree variational autoencoder (LTVAE). Whereas previous deep learning methods for clustering produce only one partition of data, LTVAE produces multiple partitions of data, each being given by one super latent variable. This is desirable because high dimensional data usually have many different natural facets and can be meaningfully partitioned in multiple ways. ",
+        "bbox": [
+            233,
+            349,
+            764,
+            503
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            536,
+            336,
+            551
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Clustering is a fundamental task in unsupervised machine learning, and it is central to many datadriven application domains. Cluster analysis partitions all the data into disjoint groups, and one can understand the structure of the data by examining examples in each group. Many clustering methods have been proposed in the literature (Aggarwal & Reddy, 2013), such as $k$ -means (MacQueen et al., 1967), Gaussian mixture models (Christopher, 2016) and spectral clustering (Von Luxburg, 2007). Conventional clustering methods are generally applied directly on the original data space. However, it is challenging to perform cluster analysis on high dimensional and unstructured data (Steinbach et al., 2004), such as images. It is not only because the dimensionality is high, but also because the original data space is too complex to interpret, e.g. there are semantic gaps between pixel values and objects in images. ",
+        "bbox": [
+            174,
+            569,
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+            708
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recently, deep learning based clustering methods have been proposed that simultanously learn nonlinear embeddings through deep neural networks and perform cluster analysis on the embedding space. The representation learning process learns effective high-level representations from high dimensional data and helps the cluster analysis. This is typically achieved by unsupervised deep learning methods, such as restricted Boltzmann machine (RBM) (Hinton et al., 2006; Hinton & Salakhutdinov, 2006), autoencoders (AE) (Vincent et al., 2008; 2010), variational autoencoders (VAE) (Kingma & Welling, 2014), etc. Previous deep learning based clustering methods (Xie et al., 2016; Guo et al., 2017; Jiang et al., 2017; Yang et al., 2017) assume one single partition over the data and that all attributes define that partition. In real-world applications, however, the assumptions are usually not true. High-dimensional data are often multifaceted and can be meaningfully partitioned in multiple ways based on subsets of attributes (Chen et al., 2012). For example, a student population can be clustered in one way based on course grades and in another way based on extracurricular activities. Movie reviews can be clustered based on both sentiment (positive or negative) and genre (comedy, action, war, etc.). It is challenging to discover the multi-facet structures of data, especially for high-dimensional data. ",
+        "bbox": [
+            174,
+            715,
+            825,
+            922
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To resolve the above issues, we propose an unsupervised learning method, latent tree variational autoencoder (LTVAE) to learn latent superstructures in variational autoencoders, and simultaneously perform representation learning and structure learning. LTVAE is a generative model, where the data is assumed to be generated from latent features through neural networks, while the latent features themselves are generated from tree-structured Bayesian networks with another level of latent variables as shown in Fig. 1. Each of those latent variables defines a facet of clustering. The proposed method automatically selects subsets of latent features for each facet, and learns the dependency structure among different facets. This is achieved through systematic structure learning. Consequently, LTVAE is able to discover complex structures of data rather than one partition. We also propose efficient learning algorithms for LTVAE with gradient descent and Stepwise EM through message passing. ",
+        "bbox": [
+            174,
+            103,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The rest of the paper is organized as follows. The related works are reviewed in Section 2. We introduce the proposed method and learning algorithms in Section 3. In Section 4, we present the empirical results. The conclusion is given in Section 5. ",
+        "bbox": [
+            176,
+            263,
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+            305
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            325,
+            352,
+            342
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Clustering has been extensively studied in the literature in many aspects (Aggarwal & Reddy, 2013). More complex clustering methods related to structure learning using Bayesian nonparametrics have been proposed, like Dirichlet Process (Blei et al., 2006), Hierarchical Dirichlet Process (HDP) (Teh et al., 2006). However, those are with conventional clustering methods that apply on raw data. Recently, deep learning based clustering methods have drawn more and more attention. A simple two-stage approach is to first learn low-dimensional embeddings using unsupervised feature learning methods, and then perform cluster analysis on the embeddings. However, without any supervision, the representation learning do not necessarily reveal the true cluster structure of the data. DEC (Xie et al., 2016) is a method that simultaneously learns feature representations and cluster assignments through deep autoencoders. It gradually improves the clustering by driving the deep network to learn a better mapping. Improved Deep Embedded Clustering (Guo et al., 2017) improves DEC by keeping the decoder network and adding reconstruction loss to the original clustering loss in DEC. Variational deep embedding (Jiang et al., 2017) is a generative method that models the data generative process using a Gaussian mixture model combined with a VAE, and also performs joint learning of representations and clustering. Similarly, GMVAE (Dilokthanakul et al., 2016) performs joint learning of a GMM and a VAE, but instead generates the mixture components through neural networks. Deep clustering network (DCN) (Yang et al., 2017) is another one that jointly learns an autoencoder and performs k-means clustering. These joint learning methods consistently achieve better clustering results than conventional ones. The method proposed in (Yang et al., 2016) uses convolutional neural networks and jointly learns the representations and clustering in a recurrent framework. All these methods assume flat partitions over the data, and do not attempt the structure learning issue. An exception is hierarchical nonparametric variational autoencoders proposed in (Goyal et al., 2017). It uses nCRP as the prior for VAE to allow infinitely deep and branching tree hierarchy structure and focuses on learning hierarchy of concepts. However, it is still one partition over the data, only that the partitions in upper levels are more general partitions, while those in lower levels more fine-grained. Different from it, our work focuses on multifacets of clustering, for example, the model could make one partition based on identity of subjects, while another partition based on pose. ",
+        "bbox": [
+            174,
+            358,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 THE PROPOSED METHOD ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            765,
+            416,
+            781
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this section, we present the proposed latent tree variational autoencoder and the learning algorithms for joint representation learning and structure learning for multidimensional clustering. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3.1 LATENT TREE VARIATIONAL AUTOENCODER ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            842,
+            524,
+            856
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Deep generative models assume that data $\\mathbf { x }$ is generated from latent continuous variable $\\mathbf { z }$ through some random process. The process consists of two steps: (1) a value $\\mathbf { z }$ is generated from some prior distribution $p ( \\mathbf { z } )$ ; (2) the observation $x$ is generated from the conditional distribution $p _ { \\boldsymbol { \\theta } } ( \\mathbf { x } | \\mathbf { z } )$ , which is parameterized through deep neural networks. Thus, it defines the joint distribution between ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/6d0b434d991986a57615a54ec0bccda8c41c57d94632b929e8edcdd1f2ac02cb.jpg",
+        "image_caption": [
+            "Figure 1: Latent Tree Variational Autoencoder "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/5fda2f4f43615c251abdb445adf97d9754edcb8a075cd2d69b2172bb8bcf51f5.jpg",
+        "image_caption": [
+            "Figure 2: Inference and gradient through message passing. Solid-arrows denote collecting message, and dashed-arrows denote distributing message. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            504,
+            102,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "observation $\\mathbf { x }$ and latent variable $\\mathbf { z }$ : ",
+        "bbox": [
+            176,
+            335,
+            406,
+            349
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/79238a8750b5905a98024572d77ef0750990387fda9e763cad0df548d5391e21.jpg",
+        "text": "$$\np ( \\mathbf { x } , \\mathbf { z } ) = p ( \\mathbf { z } ) p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            421,
+            353,
+            575,
+            371
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This process is hidden from our view, and we learn this process by maximizing the marginal loglikelihood $p ( \\mathbf { x } )$ over the parameters $\\theta$ and latent variable $\\mathbf { z }$ from data. After the learning, the latent variable $\\mathbf { z }$ can be regarded as the deep representations of $\\mathbf { x }$ since it captures the most relevant information of $\\mathbf { x }$ . Thus, the learning process is also called representation learning. ",
+        "bbox": [
+            174,
+            373,
+            825,
+            430
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In order to learn the latent structure of $\\mathbf { z }$ , for example multidimensional cluster structure, we introduce a set of latent variables $Y _ { 1 } , . . . , Y _ { l }$ on top of $\\mathbf { z }$ . A single $z$ or multiple $z$ ’s form a node $\\mathbf { z } _ { b }$ . Suppose variables in $\\mathbf { z }$ form $B$ nodes of ${ \\bf z } _ { 1 } , \\cdots , { \\bf z } _ { B }$ . Each latent variable $Y$ may be only connected to a subset of nodes, and the dependency of each $\\mathbf { z } _ { b }$ and its parent $Y$ is characterized by a conditional Gaussian distribution. Furthermore, the latent variables $Y _ { 1 } , . . . , Y _ { l }$ are connected to each other, and the dependency of a latent variable $Y$ on its parent $Y ^ { \\prime }$ is characterized by a conditional distribution $P ( Y | Y ^ { \\prime } )$ . This essentially forms a Bayesian network. And if we restrict the network to be treestructured, the $\\mathbf { z }$ and $\\mathbf { Y }$ together form a latent tree model (Zhang, 2004; Poon et al., 2010; 2013; Mourad et al., 2013; Pearl, 2014; Zhang & Poon, 2017) with $\\mathbf { z }$ being the observed variables and $\\mathbf { Y }$ being the latent variables. For multidimensional clustering, each latent variable $Y$ is taken to be a discrete variable, where each discrete state $y$ of $Y$ defines a cluster. Each latent variable $Y$ thus defines a facet partition over the data based on subset of attributes and multiple $Y$ ’s define multiple facets. Given a value $y$ of $Y$ , $\\mathbf { z } _ { b }$ follows a conditional Gaussian distribution $\\dot { P } ( \\mathbf { z } _ { b } | y ) = \\mathcal { N } ( \\mu _ { y } , \\bar { \\Sigma } _ { y } )$ with mean vector $\\mu _ { y }$ and covariance matrix $\\Sigma _ { y }$ . Thus, each $\\mathbf { z } _ { b }$ and its parent constitute a Gaussian mixture model (GMM). Suppose the parent of a node is denoted as $\\pi ( \\cdot )$ , the maginal distribution of $\\mathbf { z }$ is defined as follows ",
+        "bbox": [
+            173,
+            436,
+            825,
+            659
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3e1e56600080c12aa23c9f4da015186cb847d72633edb8fefa189a8c1f60b53f.jpg",
+        "text": "$$\np ( \\mathbf { z } ) = \\sum _ { \\mathbf { Y } } \\prod _ { j = 1 } ^ { l } p ( y _ { j } | \\pi ( Y _ { j } ) ) \\prod _ { b = 1 } ^ { B } \\mathcal { N } ( \\mathbf { z } _ { b } | \\mu _ { \\pi ( \\mathbf { z } _ { b } ) } , \\boldsymbol { \\Sigma } _ { \\pi ( \\mathbf { z } _ { b } ) } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            312,
+            660,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "which sums over all possible combinations of $\\mathbf { Y }$ states. As a matter of fact, a GMM is a Gaussian LTM that has only one latent variable connecting to all observed variables. ",
+        "bbox": [
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+            708,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Let the latent structure of $\\mathbf { Y }$ be $s$ , defining the number of latent variables in $\\mathbf { Y }$ , the number of discrete states in each variable $Y$ and the connectivity structure among all variables in $\\mathbf { z }$ and $\\mathbf { Y }$ . And let the parameters for all conditional probabilities in the latent structure be $\\Theta$ . Both the latent structure $s$ and the latent parameters $\\Theta$ are unknown. We aim to jointly learn data representations and the latent structure. The proposed LTVAE model is shown in Fig. 1. The latent structure $s$ are automatically learned from data and will be discussed in a later section. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Due to the existence of the generation network, the inference of the model is intratable. Instead, we do amortized variational inference for the latent variable $\\mathbf { z }$ by introducing an inference network (Kingma & Welling, 2014) and define an approximate posterior $q _ { \\phi } ( { \\bf z } | { \\bf x } )$ . The evidence lower bound (ELBO) $\\mathcal { L } _ { \\mathrm { E L B O } }$ of the marginal loglikelihood of the data given $( \\cal S , \\Theta )$ is: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/91c832c545217fcd8ea91d5014cf26600b68dd6c7ffa4db38f60b2d1a5663f40.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { E L B O } } ( \\mathbf { x } ) = \\mathbb { E } _ { q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) } [ \\log p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } ) ] + \\mathbb { E } _ { q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) } [ \\log \\sum _ { \\mathbf { y } } p _ { S } ( \\mathbf { z } , \\mathbf { y } ; \\Theta ) ] + \\mathbb { H } [ q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            223,
+            893,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { \\log \\sum _ { \\mathbf { y } } p _ { S } ( \\mathbf { z } , \\mathbf { y } ; \\Theta ) } \\end{array}$ is the marginal loglikelihood of the latent variable $\\mathbf { z }$ under the latent tree model, and $\\mathbb { H } [ \\cdot ]$ is the entropy. The conditional generative distribution $p _ { \\boldsymbol { \\theta } } ( \\mathbf { x } | \\mathbf { z } )$ could be a Gaussian distribution if the input data is real-valued, or a Bernoulli distribution if binary, parameterized by the generation network. Using Monte Carlo sampling, the ELBO can be asymptotically estimated by ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d05dd748160336069c8c57451e94ab4cf45caf43599216f0ab4509d06f232873.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { E L B O } } ( \\mathbf { x } ) \\simeq \\frac { 1 } { M } \\sum _ { i = 1 } ^ { M } \\log p _ { \\boldsymbol \\theta } ( \\mathbf { x } | \\mathbf { z } ^ { ( i ) } ) + \\log \\sum _ { \\mathbf { y } } p _ { \\mathcal { S } } ( \\mathbf { z } ^ { ( i ) } , \\mathbf { y } ; \\boldsymbol \\Theta ) + \\mathbb { H } [ q _ { \\boldsymbol \\phi } ( \\mathbf { z } | \\mathbf { x } ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            167,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { z } ^ { ( i ) } \\sim q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } )$ . The term $\\mathbb { H } [ q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) ]$ can be computed analytically if we choose the form of $q _ { \\phi } ( { \\bf z } | { \\bf x } )$ to be a Gaussian distribution $\\begin{array} { r } { \\mathcal { N } ( { \\bf z } ; \\mu _ { { \\bf x } } , \\sigma _ { { \\bf x } } ) \\colon \\mathbb { H } [ q _ { \\phi } ( { \\bf z } | { \\bf x } ) ] = \\frac { J } { 2 } \\log ( 2 \\pi ) + \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ { J } ( 1 + \\log \\sigma _ { j } ^ { 2 } ) . } \\end{array}$ , where $J$ is the dimensionality of $\\mathbf { z }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, the marginal loglikelihood $\\begin{array} { r } { \\log \\sum _ { \\mathbf { y } } p _ { \\mathcal { S } } ( \\mathbf { z } ^ { ( i ) } , \\mathbf { y } ; \\Theta ) } \\end{array}$ can be computed efficiently through message passing. Message passing is an efficient algorithm for inference in Bayesian networks (Koller & Friedman, 2009; Poon et al., 2013). In message passing, we first build a clique tree using the factors in the defined probability density. Because of the tree structure, each $\\mathbf { z } _ { b }$ along with its parent form a clique with the potential $\\psi ( \\mathbf { z } _ { b } , y )$ being the corresponding conditional distribution. This is illustrated in Fig. 2. With the sampled $\\mathbf { z } ^ { ( i ) }$ , we can compute the message $\\psi ^ { \\prime } ( y )$ by absorbing the evidence from $\\mathbf { z }$ . During collecting message phase, the message $\\psi ^ { \\prime } ( y )$ are sent towards the pivot. After receiving all messages, the pivot distributes back messages towards all $\\mathbf { z }$ . Both the posterior of $\\mathbf { Y }$ and the marginal loglikelihood of $\\mathbf { z } ^ { ( i ) }$ thus can be computed in the final normalization step. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 PARAMETER LEARNING THROUGH GRADIENT DESCENT AND STEPWISE EM WITH MESSAGE PASSING ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section, we propose efficient learning algorithms for LTVAE through gradient descent and stepwise EM with message passing. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Given the latent tree model $( \\cal S , \\Theta )$ , the parameters of neural networks can be efficiently optimized through stochastic gradient descent (SGD). However, in order to learn the model, it is important to efficiently compute the gradient of the marginal loglikelihood $\\log p _ { S } ( \\mathbf { z } ; \\Theta )$ from the latent tree model, the third term in Eq. 4. Here, we propose an efficient method to compute gradient through message passing. Let $\\mathbf { z } _ { b }$ be the variables that we want to compute gradient with respect to, and let $Y _ { b }$ be the parent node. The marginal loglikelihood of full $\\mathbf { z }$ can be written as ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/560bc0437868a4db77160ccf6fc1c25a8f2524617ead9402d709558fcd259688.jpg",
+        "text": "$$\n\\log p _ { \\mathcal { S } } ( \\mathbf { z } ; \\Theta ) = \\log [ \\sum _ { y _ { b } } \\mathcal { N } ( \\mathbf { z } _ { b } | \\mu _ { y _ { b } } , \\boldsymbol { \\Sigma } _ { y _ { b } } ) f ( y _ { b } ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            585,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $f ( y _ { b } )$ is the collection of all the rest of the terms not containing $\\mathbf { z } _ { b }$ . The gradient $\\mathbf { g } _ { \\mathbf { z } _ { b } }$ of the marginal loglikelihood $\\log p s ( \\mathbf { z } ; \\Theta )$ w.r.t $\\mathbf { z } _ { b }$ thus can be computed as ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6a9b51f53f467438a11f5d758c2131ec62cce654d438933acef7eab403e30615.jpg",
+        "text": "$$\n\\mathbf { g } _ { \\mathbf { z } _ { b } } = \\frac { 1 } { p _ { S } ( \\mathbf { z } ; \\boldsymbol { \\Theta } ) } \\frac { \\partial \\sum _ { y _ { b } } f ( y _ { b } ) \\mathcal { N } ( \\mathbf { z } _ { b } | \\mu _ { y _ { b } } , \\boldsymbol { \\Sigma } _ { y _ { b } } ) } { \\partial \\mathbf { z } _ { b } } = \\sum _ { y _ { b } } p ( y _ { b } | \\mathbf { z } ) \\frac { \\partial \\log [ f ( y _ { b } ) \\mathcal { N } ( \\mathbf { z } _ { b } | \\mu _ { y _ { b } } , \\boldsymbol { \\Sigma } _ { y _ { b } } ) ] } { \\partial \\mathbf { z } _ { b } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            197,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $p ( \\boldsymbol { y } _ { b } | \\mathbf { z } )$ is the posterior probability of $y _ { b }$ and can be computed efficiently with message passing as described in the previous section. The detailed derivation is in Appendix E. Since $\\mathbf { z } = [ \\bar { \\mathbf { z } _ { 1 } } , \\bar { \\mathbf { . . . , z } _ { B } } ]$ , we have ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b443f47d72f2c0f9c2ce31495e8dd99d66c94543c48aaae40fa55eb8d7394f47.jpg",
+        "text": "$$\n\\frac { \\partial \\log p ( \\mathbf { z } ) } { \\partial \\mathbf { z } } = \\left[ \\frac { \\partial \\log p ( \\mathbf { z } ) } { \\partial \\mathbf { z } _ { 1 } } , . . . , \\frac { \\partial \\log p ( \\mathbf { z } ) } { \\partial \\mathbf { z } _ { B } } \\right] = \\left[ \\mathbf { g } _ { \\mathbf { z } _ { 1 } } , . . . , \\mathbf { g } _ { \\mathbf { z } _ { B } } \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            294,
+            781,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "With the efficient computation of the third term in Eq. 4 and its gradient w.r.t $\\mathbf { z }$ through message passing, the parameters of inference network and generation network can be efficiently optimized through SGD. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In order to jointly learn the parameters of the latent tree $\\Theta$ , we propose Stepwise EM algorithm based on mini-batch of data. Specifically, we maximize the third term in Eq. 4, i.e. the marginal loglikelihood of $\\mathbf { z }$ under the latent tree. In the Stepwise E-step, we compute the distributions $\\bar { P ( } y , y ^ { \\prime } | \\mathbf { z } , \\theta ^ { ( t - 1 ) } )$ and $P ( y | \\mathbf { z } , \\theta ^ { ( t - 1 ) } )$ for each latent node $Y$ and its parent $Y ^ { \\prime }$ . In the Stepwise ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/5242bd844d379aeb381ed06e3d72093a6b1e707f80ef08bb209cd29202145d77.jpg",
+        "image_caption": [
+            "Figure 3: Structure search operators. The digits above the nodes denote the number of discrete states. Node deletion, state deletion and unpouching are the inverse of node insertion, state insertion and pouching, respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "M-step, we estimate the new parameter $\\theta ^ { ( t ) }$ . Let $\\mathbf { s } ( \\mathbf { z } , \\mathbf { y } )$ be a vector the sufficient statistics for a single data case. Let $\\bar { \\mathbf { s } } = \\mathbb { E } _ { p s ( \\mathbf { y } | \\mathbf { z } ; \\boldsymbol { \\Theta } ) } [ \\mathbf { s } ( \\mathbf { z } , \\mathbf { y } ) ]$ be the expected sufficient statistics for the data case, where the expectation is w.r.t the posterior distribution of $\\mathbf { y }$ with current parameter. And let $\\begin{array} { r } { \\boldsymbol { \\mu } = \\sum _ { i = 1 } ^ { N } \\bar { \\mathbf { s } } _ { i } } \\end{array}$ be the sum of the expected sufficient statistics. The update of the parameter $\\Theta$ is performed as follows: ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/532ba1e29945e96b04077b7285c613d009d4d528af9494d9a1672235e7a18ea7.jpg",
+        "text": "$$\n\\begin{array} { r l } & { ~ \\bar { \\mathbf { s } } _ { i } ^ { t } = \\mathbb { E } _ { p s ( \\mathbf { y } _ { i } \\mid \\mathbf { z } _ { i } ; \\boldsymbol { \\Theta } ^ { t } ) } [ \\mathbf { s } ( \\mathbf { z } _ { i } , \\mathbf { y } _ { i } ) ] } \\\\ & { ~ \\mu ^ { t + 1 } = \\mu ^ { t } + \\eta ( \\bar { \\mathbf { s } } _ { i } ^ { t } - \\mu ^ { t } ) } \\\\ & { \\boldsymbol { \\Theta } ^ { t + 1 } = \\underset { \\boldsymbol { \\Theta } } { \\arg \\operatorname* { m a x } } l ( \\mu ^ { t + 1 } , \\boldsymbol { \\Theta } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            395,
+            380,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\eta$ is the learning rate and $l$ is the complete data loglikelihood. Each iteration of update of LTVAE thus is composed of one iteration of gradient descent update for the neural network parameters and one iteration of Stepwise EM update for the latent tree model parameters with a mini-batch of data. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 STRUCTURE LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            525,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For the latent structure $s$ , there are four aspects need to determine: the number of latent variables, the cardinalities of latent variables, the connectivities among variables. We aim at finding the model $m ^ { * }$ that maximizes the BIC score (Schwarz et al., 1978; Koller & Friedman, 2009): ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/427917c00f310a6f861b77742ee1558c3ff55a864347a2ef4efb2bf4dd7e4348.jpg",
+        "text": "$$\nB I C ( m | \\mathcal { D } ) = \\log P ( \\mathcal { D } | m , \\theta ^ { * } ) - \\frac { d ( m ) } { 2 } \\log N ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            341,
+            601,
+            653,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\theta ^ { * }$ is the MLE of the parameters and $d ( m )$ is the number of independent parameters. The first term is known as the likelihood term. It favors models that fit data well. The second term is known as the penalty term. It discourages complex models. Hence, the BIC score provides a tradeoff between model fit and model complexity. To this end, we perform systematic searching to find a structure with a high BIC score. We use the hill-climing algorithm to search for $m ^ { * }$ as in (Poon et al., 2010; 2013), and define 7 search operators: node introduction (NI) and node deletion (ND) to introduce new latent nodes and delete existing nodes, state introduction (SI) and state deletion (SD) to add a new state and delete a state for existing nodes, node relocation (NR) to change links of existing nodes, pouching (PO) and unpouching (UP) operators to combine nodes into a single node and separate variables from a node.. The structure search operators are shown in Fig. 3. Each operator produces a set of candidates from existing structure, and the best candidate is picked if it improves the previous one. To reduce the number of possible search candidates, we first perform SI, NI and PO to expand the structure and pick the best model. Then we perform NR to adjust the best model. Finally, we perform UP, ND and SD to simplify the current best structure and pick the best one. Acceleration techniques (Poon et al., 2013) are adopted that make the algorithm efficient enough. The structure learning is performed iteratively together with the parameter learning of neural networks. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The overall learning algorithm is illustrated in Algorithm 1. Starting from a pretrained model, we iteratively improve the structure and parameters of latent tree model while learning the representations of data through neural network in a greedy manner. Using current structure $S ^ { t }$ as the initial structure, ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Learning Latent Tree Variational Autoencoder ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            103,
+            566,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/e9220a7dd4cbf749f234f8e825918e2c806f9bcf0272637b1ef67ab2775bd9e2.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Input:data D,z dim,neural networks,E 0,,S°,Θ° ← pretrain(D) repeat fore=1toEdo</td></tr><tr><td>for each minibatchX in D do Compute q(z|μx,Ox)</td></tr><tr><td>Sample z(i) ~q(z|μx,x) Compute log ps(z; ) and 0log ps(z) from Eq.5 and 7 dz</td></tr><tr><td>Compute ELBO from Eq. 4 0,Φ ← Back-propagation and SGD step</td></tr><tr><td>← StepwiseEM(z()) end for</td></tr><tr><td>end for Dz↑μD</td></tr><tr><td></td></tr><tr><td></td></tr><tr><td>repeat</td></tr><tr><td>S*,Θ* ← SearchWith(St-1,Θt-1,{SI,NI,PO})</td></tr><tr><td>S*,Θ* ← SearchWith(S*,Θ* {NR}) St,Θt ← SearchWith(S*,Θ*,{UP,ND,SD}) until BIC(St,0t|Dz)≤BIC(St-1,0t-1|Dz)</td></tr></table>",
+        "bbox": [
+            179,
+            122,
+            568,
+            397
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "we search for a better model. With new latent tree model, we optimize for a better representation until convergence. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            473,
+            326,
+            488
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 SYNTHETIC-DATA DEMONSTRATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            503,
+            467,
+            518
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We first demonstrate the effectiveness of the proposed method through synthetic data. Assume that the data points have two facets $Y _ { 1 }$ and $Y _ { 2 }$ , where each facet controlls a subset of attributes (e.g. two-dimensional domain) and defines one partition over the data. This four-dimensional domain $\\mathbf { z } = \\{ z _ { 1 } , z _ { 2 } , z _ { 3 } , z _ { 4 } \\}$ is a latent representation which we do not observe. What we observe is $\\mathbf { x } \\in \\mathbb { R } ^ { 1 0 0 }$ that is obtained via the following non-linear transformation: ",
+        "bbox": [
+            173,
+            530,
+            825,
+            599
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/55cabb713797d6a5aa299c05109bc80c64656e1ac007094be6ea43c60aaebc9b.jpg",
+        "text": "$$\n\\mathbf { x } = \\sigma ( U \\sigma ( W \\mathbf { z } ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            436,
+            607,
+            560,
+            625
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $W \\in \\mathbb { R } ^ { 1 0 \\times 4 }$ and $U \\in \\mathbb { R } ^ { 1 0 0 \\times 1 0 }$ are matrices whose entries follow the zero-mean unit-variance i.i.d. Gaussian distribution, $\\sigma ( \\cdot )$ is a sigmoid function to introduce nonlinearity. The generative model is shown in Fig. 4 (a). We define two clusters in facet $Y _ { 1 }$ and two clusters in facet $Y _ { 2 }$ , and generate 5,000 samples of $\\mathbf { x }$ . Under the above generative model, recovering the two facets $Y _ { 1 }$ and $Y _ { 2 }$ structure and the latent $z$ domain from the observation of $\\mathbf { x }$ seems very challenging. All previous DNN-based methods $\\mathbf { \\Gamma } ( \\mathbf { A E + G M M }$ , DEC, DCN, etc.) are only able to discover one-facet of clustering (i.e. one partition over the data), and none of these is applicable to solve such a multidimensional clustering problem. Fig. 4 (b) shows the results of the proposed method. As one can see, the LTVAE successfully discovers the true superstructure of $Y _ { 1 }$ and $Y _ { 2 }$ . The 2-d plot of $z _ { 1 }$ and $z _ { 2 }$ shows the separable latent space clusters under facet $Y _ { 1 }$ , and it matches the ground-truth cluster assignments. Additionally, the 2-d plot of $z _ { 3 }$ and $z _ { 4 }$ shows another separable clusters under facet $Y _ { 2 }$ , and it also matches the ground-truth cluster assignments well in the other facet. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 REAL-DATA EXPERIMENT SETUP ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "We evaluate the proposed LTVAE model on two image datasets and two other datasets, and compare it against other deep learning based clustering algorithms, including two-stage methods, $\\mathbf { A E { + } G M M }$ and $\\mathbf { V A E + G M M }$ , which first learn AE/VAE (Kingma & Welling, 2014) models then construct a GMM on top of them, and joint learning methods, DEC (Xie et al., 2016) and DCN (Yang et al., 2017). The datasets include MNIST, STL-10, Reuters (Xie et al., 2016; Jiang et al., 2017) and the Heterogeneity Human Activity Recognition (HHAR) dataset. When evaluating the clustering performance, for fair of comparison, we follow previous works (Xie et al., 2016; Yang et al., 2017) and use the network structures of $d - 5 0 0 - 5 0 0 - 2 0 0 0 - 1 0$ for the encoder network and $1 0 - 2 0 0 0 -$ $5 0 0 - 5 0 0 - d$ for the decoder network for all datasets, where $d$ is the data-space dimension, which varies among datasets. All layers are fully-connected. We follow the pretraining procedure as in (Xie et al., 2016). We first perform greedy layer-wise pretraining in denoising autoencoder manner, then stack all layers to form deep autoencoder. The deep autoencoder is further finetuned to minimize the reconstruction loss. The weights of the deep autoencoder are used to intialize the weights of encoder and decoder networks of above methods. After the pretraining, we optimze the objectives of those methods. For DEC and DCN, we use the same hyperparameter settings as the original papers. When initializing the cluster centroids for DEC and DCN, we perform 10 random restarts and pick the results with the best objective value for $k$ -means/GMM. For the proposed LTVAE, we use Adam optimzer (Kingma & Ba, 2015) with initial learning rate of 0.001 and mini-batch size of 128. For Stepwise EM, we set the learning rate to be 0.01. As in Algorithm 1, we set $E = 5$ , i.e. we update the latent tree model every 5 epochs. When optimizing the candidate models during structure search, we perform 10 random restarts and train with EM for 200 iterations. ",
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+        "img_path": "images/e672589b914b2a28cffd51a0a4a0d6081efff6a689a04a75ded1c725ef434333.jpg",
+        "image_caption": [
+            "Figure 4: (a) The generative process of synthetic data; (b) The discovered multidimensional superstructure and the latent space (different colors denote different ground truth clusters in each facet.) "
+        ],
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+        "table_caption": [
+            "Table 1: Test data loglikelihood for various datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>MNIST</td><td rowspan=1 colspan=1>STL</td><td rowspan=1 colspan=1>Reuters</td><td rowspan=1 colspan=1>HHAR</td></tr><tr><td rowspan=1 colspan=1>VAE</td><td rowspan=1 colspan=1>-86.64±0.20</td><td rowspan=1 colspan=1>-743.72±0.51</td><td rowspan=1 colspan=1>-1312.40±1.24</td><td rowspan=1 colspan=1>-17.88±0.26</td></tr><tr><td rowspan=1 colspan=1>IWAE</td><td rowspan=1 colspan=1>-85.39±0.13</td><td rowspan=1 colspan=1>-742.43±1.10</td><td rowspan=1 colspan=1>-1254.29±6.95</td><td rowspan=1 colspan=1>-16.53±0.16</td></tr><tr><td rowspan=1 colspan=1>LTVAE</td><td rowspan=1 colspan=1>-84.75±0.14</td><td rowspan=1 colspan=1>-619.04±7.88</td><td rowspan=1 colspan=1>-1245.71±4.45</td><td rowspan=1 colspan=1>-13.65±0.53</td></tr></table>",
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+        "text": "4.3 TEST LOGLIKELIHOOD ",
+        "text_level": 1,
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+        "text": "We first show that, by using the marginal loglikelihood defined by the latent tree model as the prior, LTVAE better fits the data than conventional VAE and importance weighted autoencoders (IWAE) (Burda et al., 2016). While alternative quantitative criteria have been proposed (Bounliphone et al., 2016; Im et al., 2016; Salimans et al., 2016) for generative models, log-likelihood of held-out test data remains one of the most important measures of a generative model’s performance (Kingma & Welling, 2014; Burda et al., 2016; Wu et al., 2017; Goyal et al., 2017). For comparison, we approximate true loglikelihood $\\mathcal { L } _ { 5 0 0 0 }$ using importance sampling (Burda et al., 2016): ${ \\mathcal { L } } _ { k } ( \\mathbf { x } ) =$ $\\begin{array} { r } { \\log { \\frac { 1 } { k } } \\sum _ { i = 1 } ^ { k } \\frac { p _ { \\theta } ( \\mathbf { x } , \\mathbf { z } ^ { ( i ) } ) } { q _ { \\phi } ( \\mathbf { z } ^ { ( i ) } | \\mathbf { x } ) } } \\end{array}$ , where $\\mathbf { z } ^ { ( i ) } \\sim q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } )$ . The results for all datasets are shown in Table 1. The proposed LTVAE obtains a higher test data loglikelihood and ELBO, implying that it can better model the underlying complex data distribution embedded in the image data. ",
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+        "img_path": "images/1d233589357b0631e59d50b485f75fabf632ff9d82873f460dacc04ee477949a.jpg",
+        "image_caption": [
+            "Figure 5: Two facet clustering results from LTVAE are in (a) digit identity and (b) shape and pose. Each row contains the top 10 scoring elements from one cluster. (c) shows the pose variations by fixing the digit cluster in facet 1 and changing the cluster in facet 2. It can be seen that up-right, left-tilted and right-tilted images of the same digits are clearly recognizable. "
+        ],
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+        "table_caption": [
+            "Table 2: Clustering Accuracy of clustering results. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>MNIST</td><td rowspan=1 colspan=1>STL-10</td><td rowspan=1 colspan=1>Reuters</td><td rowspan=1 colspan=1>HHAR</td></tr><tr><td rowspan=1 colspan=1>AE+GMM</td><td rowspan=1 colspan=1>82.18%</td><td rowspan=1 colspan=1>79.83%</td><td rowspan=1 colspan=1>68.68%</td><td rowspan=1 colspan=1>78.90%</td></tr><tr><td rowspan=1 colspan=1>VAE+GMM</td><td rowspan=1 colspan=1>76.87%</td><td rowspan=1 colspan=1>79.49%</td><td rowspan=1 colspan=1>65.85%</td><td rowspan=1 colspan=1>67.91%</td></tr><tr><td rowspan=1 colspan=1>DEC</td><td rowspan=1 colspan=1>84.30%</td><td rowspan=1 colspan=1>80.62%</td><td rowspan=1 colspan=1>74.32%</td><td rowspan=1 colspan=1>79.86%</td></tr><tr><td rowspan=1 colspan=1>DCN</td><td rowspan=1 colspan=1>83.32%</td><td rowspan=1 colspan=1>85.88%</td><td rowspan=1 colspan=1>75.05%</td><td rowspan=1 colspan=1>81.26%</td></tr><tr><td rowspan=1 colspan=1>LTVAE</td><td rowspan=1 colspan=1>86.32%</td><td rowspan=1 colspan=1>90.00%</td><td rowspan=1 colspan=1>80.96%</td><td rowspan=1 colspan=1>85.00%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.4 MULTIFACET CLUSTERING ",
+        "text_level": 1,
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+        "text": "The most important features of the proposed model are that it can perform variable selection for model-based clustering, leading to multiple facets clustering. ",
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+        "type": "text",
+        "text": "We use the standard unsupervised evaluation metric and protocols for evaluations and comparisons to other algorithms (Yang et al., 2010). For baseline algorithms we set the number of clusters to the number of ground-truth categories. While for LTVAE, it automatically determines the number of facets and latent superstructure through structure learning. We evaluate performance with unsupervised clustering accuracy $( A C C )$ : ",
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+    {
+        "type": "equation",
+        "img_path": "images/d30bbbb2230c205c6232f763d2eb26f2b822f7f84991d08f73243793394042b4.jpg",
+        "text": "$$\nA C C = \\operatorname* { m a x } _ { m } \\frac { \\sum _ { i = 1 } ^ { n } \\mathbf { 1 } \\left\\{ l _ { i } = m ( c _ { i } ) \\right\\} } { n } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $l _ { i }$ is the groundtruth label, $c _ { i }$ is the cluster assignment produced by the algorithm, and $m$ ranges over all possible mappings between clusters and labels. Table 2 show the quantitative clustering results compared with previous works. With $\\mathbf { z }$ dimension of small value like 10, LTVAE usually discovers only one facet. It can be seen the, for MNIST dataset LTVAE achieves clustering accuracy of $8 6 . 3 2 \\%$ , better than the results of other methods. This is also the case for STL-10, Reuters and HHAR. ",
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+    {
+        "type": "text",
+        "text": "More importantly, the proposed LTVAE does not just give one partition over the data. Instead, it explains the data in multi-faceted ways. Unlike previous clustering experiments, for this experiment, we choose the z dimension to be 20. Fig. 5 shows the two facet clustering results for MNIST. It can be seen that facet 1 gives quite clean clustering over the identity of the digits and the ten digits are well separated. On the other hand, facet 2 gives a more grand partition based on the shape and pose. Note how up-right $\" 4 > \"$ and “9” are similar, and how tilted “4”,“7” and “9” are similar. The facet meanings are more evident in Fig. 5 (c). Fig. 6 shows four facets discovered for the STL-10 dataset. Although it is hard to characterize precisely how the facets differ from each other, there are visible patterns. For example, the cats, monkeys and birds in facet 2 have clearly visible eyes, while this is not always true in facet 1. The deers in facet 2 are all showing their antlers/ears, while this is not true in facet 3. In facet 2 we see frontal views of cars, while in facets 1 and 3 we see side view of cars. In facet 1, each cluster consists of the same types of objects/animal. In facet 3/4, images in the same cluster do not necessarily show the same type of objects/animals. However, they have similar overall feel. ",
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+    {
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+        "img_path": "images/ff99c4bc8ecd2f35cfc249876f70070a5f186e93efe69e4180f070a4bf2f59d6.jpg",
+        "image_caption": [
+            "Figure 6: Clustering results from LTVAE for STL-10 dataset. Each row contains the top 5 scoring elements from one cluster. "
+        ],
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+    },
+    {
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+        "img_path": "images/c09285f05ed4a5f8198b024ddb48f70e4a9e48fdbbb909c8d2925f06a53057f9.jpg",
+        "image_caption": [
+            "Figure 7: The digits generated by the proposed model. Digits in the same row come from the same latent code of the latent tree. "
+        ],
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+    {
+        "type": "text",
+        "text": "4.5 IMAGE GENERATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Since the structure of the data in latent space is automatically learned through the latent tree, we can sample the data in a more structured way. One way is through ancestral sampling, where we first sample the root of the latent tree and then hierarchically sample the children variables to get $\\mathbf { z }$ , from which the images can be generated through generation network. The other way is to pick one component from the Gaussian mixture and sample $\\mathbf { z }$ from that component. This produces samples from a particular cluster. Fig. 7 shows the samples generated in this way. As it can be seen, digits sampled from each component has clear semantic meaning and belong to the same category. Whereas, the samples generated by VAE does not have such structure. Conditional image generation can also be performed to alter the attributes of the same digit as shown in Appendix B. ",
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+    {
+        "type": "text",
+        "text": "5 DISCUSSIONS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "LTVAE learns the dependencies among latent variables $\\mathbf { Y }$ . In general, latent variables are often correlated. For example, the social skills and academic skills of a student are generally correlated. Therefore, its better to model this relationship to better fit the data. Experiments show that removing such dependencies in LTVAE models results in inferior data loglikelihood. ",
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+    {
+        "type": "text",
+        "text": "In this paper, for the inference network, we simply use mean-field inference network with same structure as the generative network (Kingma & Welling, 2014). However, the limited expressiveness of the mean-field inference network could restrict the learning in the generative network and the quality of the learned model (Webb et al., 2018; Rainforth et al., 2018; Cremer et al., 2018). Using a faithful inference network structure as in (Webb et al., 2018) to incorporate the dependencies among latent variables in the posterior, for example one parameterized with masked autoencoder distribution estimator (MADE) model (Germain et al., 2015), could have a significant improvement in learning. We leave it for future investigation. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose an unsupervised learning method, latent tree variational autoencoder (LTVAE), which simultaneously performs representation learning and multidimensional clustering. Different from previous deep learning based clustering methods, LTVAE learns latent embeddings from data and discovers multi-facet clustering structure based on subsets of latent features rather than one partition over data. Experiments show that the proposed method achieves state-of-the-art clustering performance and reals reasonable multifacet structures of the data. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Research on this article was supported by Hong Kong Research Grants Council under grants 16212516 and 16202118. ",
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+        "text": "Nevin L Zhang. Hierarchical latent class models for cluster analysis. Journal of Machine Learning Research, 5(6):697–723, 2004. ",
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+    {
+        "type": "text",
+        "text": "Nevin L Zhang and Leonard KM Poon. Latent tree analysis. In AAAI, pp. 4891–4898, 2017. ",
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+            169,
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A SUPERSTRUCTURES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            375,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For the MNIST dataset, the conditional probability between identity facet $Y _ { 1 }$ ( $\\mathbf { \\bar { X } } ^ { \\prime }$ -axis) and pose facet $Y _ { 2 }$ (y-axis) is shown in Fig. 8. It can be seen that a cluster in $Y _ { 1 }$ facet could correspond to multiple clusters in $Y _ { 2 }$ facet due to the conditional probability, e.g. cluster 0, 4, 5, 11 and 12. However, not all clusters in $Y _ { 2 }$ facet are possible for a given cluster in $Y _ { 1 }$ facet. ",
+        "bbox": [
+            173,
+            137,
+            826,
+            194
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/d997c4363a0c39b11d777cc78d967c979fc2920e26ce9822474c2bcb0761d59c.jpg",
+        "image_caption": [
+            "Figure 8: Conditional probability of $Y _ { 1 }$ and $Y _ { 2 }$ for the two facets of MNIST discovered by LTVAE. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            336,
+            236,
+            638,
+            415
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B CONDITIONAL IMAGE GENERATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            501,
+            504,
+            517
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Here we show more results on conditional image generation. Interestingly, with LTVAE, we can change the original images by fixing variables in some facet and sampling in other facets. For example, in MNIST we can fix the variables in identity facet and change the pose of the digit by sampling in the pose facet. Fig. 9 shows the samples generated in this way. As it can be seen, the pose of the input digits are changed in the samples generated by the proposed method. ",
+        "bbox": [
+            173,
+            536,
+            825,
+            608
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/305531c7e4bf0fba99c111714fa1b632f5f0caa852d4e3492a11f51cc6df7b31.jpg",
+        "image_caption": [
+            "Figure 9: Image generation. Left are the original image. Right are generated with the proposed model by fixing the variables in identity facet and sampling the variables in the pose facet. Digits in the same row come from the same latent code of the latent tree. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            333,
+            626,
+            663,
+            844
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C COMPUTATIONAL TIME ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            403,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We compare the computational time of the proposed LTVAE w/ structure learning and that w/ fixed structure. For LTVAE with fixed structure, we fixed the structure of the latent tree model to be a single $Y$ connecting to all $z \\mathbf { S }$ , in which each $z$ node consists of single $z$ variable. ",
+        "bbox": [
+            174,
+            142,
+            825,
+            185
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/6900b7b87219b515878923bc091c63076363457f37c26806084c5ee457cab0e4.jpg",
+        "table_caption": [
+            "Table 3: Computational time (s) w/ and w/o structure learning. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>MNIST</td><td rowspan=1 colspan=1>STL-10</td><td rowspan=1 colspan=1>Reuters</td><td rowspan=1 colspan=1>HHAR</td></tr><tr><td rowspan=1 colspan=1>LTVAE w/ structure learning</td><td rowspan=1 colspan=1>7,592</td><td rowspan=1 colspan=1>3,251</td><td rowspan=1 colspan=1>5,756</td><td rowspan=1 colspan=1>5,693</td></tr><tr><td rowspan=1 colspan=1>LTVAE w/ fixed structure</td><td rowspan=1 colspan=1>3,197</td><td rowspan=1 colspan=1>542</td><td rowspan=1 colspan=1>1,442</td><td rowspan=1 colspan=1>1,021</td></tr></table>",
+        "bbox": [
+            253,
+            243,
+            741,
+            292
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D SIMILARITY BETWEEN LEARNED FACETS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            361,
+            552,
+            377
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Different facets learned by LTVAE might have some overlaps among each other. Here we make quantitative comparison among different facets based on the cluster assignments in each facet. We evaluate the similarity between two clusterings $Y _ { 1 }$ and $Y _ { 2 }$ using normalized mutual information $N M I ( Y _ { 1 } ; Y _ { 2 } )$ . The NMI is given by ",
+        "bbox": [
+            173,
+            401,
+            825,
+            458
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/67ce79d04e818f6b17f352a01f8a06857659a6bbd34ddd0e306fa8640fc1cbc8.jpg",
+        "text": "$$\nN M I ( Y _ { 1 } ; Y _ { 2 } ) = \\frac { I ( Y _ { 1 } ; Y _ { 2 } ) } { \\sqrt { H ( Y _ { 1 } ) H ( Y _ { 2 } ) } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            477,
+            614,
+            515
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where $I ( Y _ { 1 } ; Y _ { 2 } )$ is the mutual information between $Y _ { 1 }$ and $Y _ { 2 }$ and $H ( V )$ is the entropy of a variable $V$ . These quantities can be computed from $P ( Y _ { 1 } ; Y _ { 2 } )$ , which in turn is estimated by $P ( Y _ { 1 } ; Y _ { 2 } ) =$ $\\begin{array} { r l } { { } } & { { } { \\frac { 1 } { N } } \\sum _ { i = 1 } ^ { N } P ( Y _ { 1 } | \\mathbf { d _ { i } } ) P ( Y _ { 2 } | \\mathbf { d _ { i } } ) } \\end{array}$ , where $\\mathbf { d } _ { 1 } , \\cdots , \\mathbf { d } _ { N }$ are the samples in the test data. ",
+        "bbox": [
+            174,
+            534,
+            825,
+            582
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/298a0f16edab03a5234a57c316f0488ee9cbe8279dd04b2004b500902cbb1cda.jpg",
+        "table_caption": [
+            "Table 4: NMI between different facets learned by LTVAE and groundtruth for MNIST dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Groundtruth</td><td rowspan=1 colspan=1>Facet 1</td><td rowspan=1 colspan=1>Facet 2</td></tr><tr><td rowspan=1 colspan=1>Groundtruth</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0.825</td><td rowspan=1 colspan=1>0.574</td></tr><tr><td rowspan=1 colspan=1>Facet 1</td><td rowspan=1 colspan=1>0.825</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0.682</td></tr><tr><td rowspan=1 colspan=1>Facet 2</td><td rowspan=1 colspan=1>0.574</td><td rowspan=1 colspan=1>0.682</td><td rowspan=1 colspan=1>1</td></tr></table>",
+        "bbox": [
+            330,
+            640,
+            666,
+            703
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/e0db474f052c0c01c414682ca3fb50ca2934edf0af0b04938fd8191272e711e8.jpg",
+        "table_caption": [
+            "Table 5: NMI between different facets learned by LTVAE and groundtruth for STL dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Groundtruth</td><td rowspan=1 colspan=1>Facet1</td><td rowspan=1 colspan=1>Facet 2</td><td rowspan=1 colspan=1>Facet3</td><td rowspan=1 colspan=1>Facet 4</td></tr><tr><td rowspan=1 colspan=1>Groundtruth</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0.8613</td><td rowspan=1 colspan=1>0.6279</td><td rowspan=1 colspan=1>0.5758</td><td rowspan=1 colspan=1>0.5536</td></tr><tr><td rowspan=1 colspan=1>Facet 1</td><td rowspan=1 colspan=1>0.8613</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0.6962</td><td rowspan=1 colspan=1>0.6314</td><td rowspan=1 colspan=1>0.6251</td></tr><tr><td rowspan=1 colspan=1>Facet 2</td><td rowspan=1 colspan=1>0.6279</td><td rowspan=1 colspan=1>0.6962</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0.4675</td><td rowspan=1 colspan=1>0.5031</td></tr><tr><td rowspan=1 colspan=1>Facet 3</td><td rowspan=1 colspan=1>0.5758</td><td rowspan=1 colspan=1>0.6314</td><td rowspan=1 colspan=1>0.4675</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0.5885</td></tr><tr><td rowspan=1 colspan=1>Facet 4</td><td rowspan=1 colspan=1>0.5536</td><td rowspan=1 colspan=1>0.6251</td><td rowspan=1 colspan=1>0.5031</td><td rowspan=1 colspan=1>0.5885</td><td rowspan=1 colspan=1>1</td></tr></table>",
+        "bbox": [
+            263,
+            795,
+            735,
+            887
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "E DERIVATION OF GRADIENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            431,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Here we give detailed derivation of Equation 6. The gradient $\\mathbf { g } _ { \\mathbf { z } _ { b } }$ of the marginal loglikelihood $\\log p s ( { \\bf z } ; \\Theta )$ w.r.t $\\mathbf { z } _ { b }$ thus can be computed as ",
+        "bbox": [
+            173,
+            132,
+            823,
+            162
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/af712af15f2b252b1df6409f962cc79436bd1e2cf36290e0cf0d5e9f7aeb84a9.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\mathcal { Q } _ { \\kappa , \\kappa } = } & { \\frac { \\partial \\langle \\Phi _ { \\kappa } | \\phi _ { \\sigma } \\rangle \\langle \\kappa \\Theta \\rangle } { \\partial \\mathcal { L } _ { \\kappa } } } \\\\ & { = \\frac { \\partial } { \\partial \\kappa _ { \\kappa } \\langle \\kappa \\Theta \\rangle } _ { \\kappa } \\frac { \\partial \\langle \\Phi _ { \\kappa } | \\phi _ { \\sigma } \\rangle _ { \\kappa } } { \\partial \\mathcal { L } _ { \\kappa } } } \\\\ & { = \\frac { 1 } { \\rho _ { \\kappa } \\langle \\kappa \\Theta \\rangle } _ { \\kappa } \\frac { \\partial \\sum _ { \\kappa } f \\langle \\Phi _ { \\kappa } \\rangle \\langle \\kappa \\Theta _ { \\kappa } | \\phi _ { \\kappa } \\rangle _ { \\kappa } \\langle \\kappa \\Theta _ { \\kappa } \\rangle } { \\partial \\mathcal { L } _ { \\kappa } } } \\\\ & { = \\sum _ { \\kappa } \\frac { \\partial } { \\rho _ { \\kappa } \\langle \\kappa \\Theta \\rangle } _ { \\kappa } \\frac { \\partial | f \\langle \\Phi _ { \\kappa } \\rangle \\langle \\kappa | \\Theta _ { \\kappa } \\rangle _ { \\kappa } \\langle \\kappa \\Theta _ { \\kappa } \\rangle | } { \\partial \\mathcal { L } _ { \\kappa } } } \\\\ & { = \\sum _ { \\kappa } \\frac { f \\langle \\Phi _ { \\kappa } | \\Phi \\rangle \\langle \\kappa | \\Phi _ { \\kappa } | \\rho _ { \\kappa } \\rangle _ { \\kappa } \\langle \\kappa \\Theta _ { \\kappa } | \\phi _ { \\kappa } \\rangle _ { \\kappa } } { \\partial \\mathcal { L } _ { \\kappa } \\langle \\kappa \\Theta \\rangle } \\frac { \\partial | f \\langle \\Phi _ { \\kappa } | \\mathcal { N } \\langle \\kappa | \\Phi _ { \\kappa } \\rangle _ { \\kappa } \\langle \\kappa \\Theta _ { \\kappa } \\rangle _ { \\kappa } | } { \\partial \\mathcal { L } _ { \\kappa } } } \\\\ & { = \\sum _ { \\kappa } \\frac { f \\langle \\Phi _ { \\kappa } | \\Phi \\rangle \\langle \\kappa | \\Phi _ { \\kappa } | \\rho _ { \\kappa } \\rangle _ { \\kappa } \\langle \\kappa | \\Phi _ { \\kappa } \\rangle _ { \\kappa } \\langle \\kappa | \\Phi _ { \\kappa } \\rangle _ { \\kappa } } { \\partial \\mathcal { L } _ { \\kappa } } } \\\\ & { = \\sum _ { \\kappa } \\rho _ { \\kappa } | z | \\rho _ { \\kappa } \\langle | \\Phi _ { \\kappa } | \\rho _ { \\kappa } \\rangle _ { \\kappa } \\langle | \\Phi _ { \\kappa } | \\rho _ { \\kappa } \\rangle _ { \\kappa } } \\\\ &  = \\sum _ { \\kappa } ^ { \\kappa } \\rho _ { \\kappa } | z | \\rho _ { \\kappa } \\langle \\kappa | \\Phi _ { \\kappa } \\rangle _ { \\kappa }  \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            292,
+            169,
+            700,
+            433
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "where $p ( \\boldsymbol { y } _ { b } | \\mathbf { z } )$ is the posterior probability of $y _ { b }$ and can be computed efficiently with message passing as described in the previous section. Note that ",
+        "bbox": [
+            174,
+            436,
+            821,
+            465
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3c65a06811bc1f19192f3bbb52c88209382a455a27323efb56c068d8766384a7.jpg",
+        "text": "$$\np ( y _ { b } | \\mathbf { z } ) = \\frac { f ( y _ { b } ) \\mathcal { N } ( \\mathbf { z } _ { b } | \\mu _ { y _ { b } } , \\Sigma _ { y _ { b } } ) } { p s ( \\mathbf { z } ; \\boldsymbol { \\Theta } ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            470,
+            607,
+            506
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "is valid due to $\\begin{array} { r } { p _ { S } ( \\mathbf { z } ; \\Theta ) = \\sum _ { y _ { b } } \\mathcal { N } ( \\mathbf { z } _ { b } | \\mu _ { y _ { b } } , \\boldsymbol { \\Sigma } _ { y _ { b } } ) f ( y _ { b } ) } \\end{array}$ and the Bayes rule. ",
+        "bbox": [
+            173,
+            511,
+            655,
+            531
+        ],
+        "page_idx": 14
+    }
+]

parse/train/hl3v8io3ZYt/hl3v8io3ZYt.md

@@ -0,0 +1,268 @@
+# Associating Objects with Transformers for Video Object Segmentation
+
+Zongxin Yang1,2, Yunchao Wei3,4, Yi Yang1
+
+1 CCAI, College of Computer Science and Technology, Zhejiang University 2 Baidu Research 3 Institute of Information Science, Beijing Jiaotong University 4 Beijing Key Laboratory of Advanced Information Science and Network {zongxinyang1996, wychao1987, yee.i.yang}@gmail.com
+
+# Abstract
+
+This paper investigates how to realize better and more efficient embedding learning to tackle the semi-supervised video object segmentation under challenging multi-object scenarios. The state-of-the-art methods learn to decode features with a single positive object and thus have to match and segment each target separately under multi-object scenarios, consuming multiple times computing resources. To solve the problem, we propose an Associating Objects with Transformers (AOT) approach to match and decode multiple objects uniformly. In detail, AOT employs an identification mechanism to associate multiple targets into the same high-dimensional embedding space. Thus, we can simultaneously process multiple objects’ matching and segmentation decoding as efficiently as processing a single object. For sufficiently modeling multi-object association, a Long Short-Term Transformer is designed for constructing hierarchical matching and propagation. We conduct extensive experiments on both multi-object and single-object benchmarks to examine AOT variant networks with different complexities. Particularly, our R50-AOT-L outperforms all the state-of-the-art competitors on three popular benchmarks, i.e., YouTube-VOS $( 8 4 . 1 \% ~ \mathcal { I } \& \mathcal { F } )$ , DAVIS 2017 $( 8 4 . 9 \% )$ , and DAVIS 2016 $( 9 1 . 1 \% )$ , while keeping more than $3 \times$ faster multi-object run-time. Meanwhile, our AOT-T can maintain real-time multi-object speed on the above benchmarks. Based on AOT, we ranked $\mathbf { 1 ^ { s t } }$ in the 3rd Large-scale VOS Challenge.
+
+# 1 Introduction
+
+Video Object Segmentation (VOS) is a fundamental task in video understanding with many potential applications, including augmented reality [25] and self-driving cars [52]. The goal of semi-supervised VOS, the main task in this paper, is to track and segment object(s) across an entire video sequence based on the object mask(s) given at the first frame.
+
+Thanks to the recent advance of deep neural networks, many deep learning based VOS algorithms have been proposed recently and achieved promising performance. STM [26] and its following works [34, 23] leverage a memory network to store and read the target features of predicted past frames and apply a non-local attention mechanism to match the target in the current frame. FEELVOS [41] and CFBI [50, 51] utilize global and local matching mechanisms to match target pixels or patches from both the first and the previous frames to the current frame.
+
+Even though the above methods have achieved significant progress, the above methods learn to decode scene features that contain a single positive object. Thus under a multi-object scenario, they have to match each object independently and ensemble all the single-object predictions into a multi-object segmentation, as shown in Fig. 1a. Such a post-ensemble manner eases network architectures’ design since the networks are not required to adapt the parameters or structures for different object numbers. However, modeling multiple objects independently, instead of uniformly, is inefficient in exploring multi-object contextual information to learn a more robust feature representation for VOS. In addition, processing multiple objects separately yet in parallel requires multiple times the amount of GPU memory and computation for processing a single object. This problem restricts the training and application of VOS under multi-object scenarios, especially when computing resources are limited.
+
+![](images/4b20923d50ad97a87da9b6849c8352dfcb5e2697e5936a15d3f25e62b6e5b068.jpg)  
+Figure 1: VOS methods (e.g., [50, 34]) process multi-object scenarios in a post-ensemble manner (a). In contrast, our AOT associates all the objects uniformly (b), leading to better efficiency (c).
+
+To solve the problem, Fig. 1b demonstrates a feasible approach to associate and decode multiple objects uniformly in an end-to-end framework. Hence, we propose an Associating Objects with Transformers (AOT) approach to match and decode multiple targets uniformly. First, an identification mechanism is proposed to assign each target a unique identity and embed multiple targets into the same feature space. Hence, the network can learn the association or correlation among all the targets. Moreover, the multi-object segmentation can be directly decoded by utilizing assigned identity information. Second, a Long Short-Term Transformer (LSTT) is designed for constructing hierarchical object matching and propagation. Each LSTT block utilizes a long-term attention for matching with the first frame’s embedding and a short-term attention for matching with several nearby frames’ embeddings. Compared to the methods [26, 34] utilizing only one attention layer, we found hierarchical attention structures are more effective in associating multiple objects.
+
+We conduct extensive experiments on two popular multi-object benchmarks for VOS, i.e., YouTubeVOS [48] and DAVIS 2017 [31], to validate the effectiveness and efficiency of the proposed AOT. Even using the light-weight Mobilenet-V2 [33] as the backbone encoder, the AOT variant networks achieve superior performance on the validation 2018 & 2019 splits of the large-scale YouTube-VOS (ours, $\mathcal { T } \& \mathcal { F } 8 2 . 6 \sim 8 4 . 5 \%$ & $8 2 . 2 \sim 8 4 . 5 \%$ while keeping more than $\mathbf { 2 \times }$ faster multi-object run-time $( \mathbf { 2 7 . 1 } \sim \mathbf { 9 . 3 F P S }$ ) compared to the state-of-the-art competitors (e.g., CFBI [50], $8 1 . 4 \%$ & $8 1 . 0 \%$ , 3.4FPS). We also achieve new state-of-the-art performance on both the DAVIS-2017 validation $( 8 5 . 4 \% )$ and testing $( 8 1 . 2 \% )$ splits. Moreover, AOT is effective under single-object scenarios as well and outperforms previous methods on DAVIS 2016 [30] $( 9 2 . 0 \% )$ , a popular single-object benchmark. Besides, our smallest variant, AOT-T, can maintain real-time multi-object speed on all above benchmarks (51.4FPS on $4 8 0 \mathrm { p }$ videos). Particularly, AOT ranked $\mathbf { 1 ^ { s t } }$ in the Track 1 (Video Object Segmentation) of the 3rd Large-scale Video Object Segmentation Challenge.
+
+Overall, our contributions are summarized as follows:
+
+• We propose an identification mechanism to associate and decode multiple targets uniformly for VOS. For the first time, multi-object training and inference can be efficient as single-object ones, as demonstrated in Fig. 1c. • Based on the identification mechanism, we design a new efficient VOS framework, i.e., Long ShortTerm Transformer (LSTT), for constructing hierarchical multi-object matching and propagation. LSTT achieves superior performance on VOS benchmarks [48, 31, 30] while maintaining better efficiency than previous state-of-the-art methods. To the best of our knowledge, LSTT is the first hierarchical framework for object matching and propagation by applying transformers [39] to VOS.
+
+# 2 Related Work
+
+Semi-supervised Video Object Segmentation. Given one or more annotated frames (the first frame in general), semi-supervised VOS methods propagate the manual labeling to the entire video sequence. Traditional methods often solve an optimization problem with an energy defined over a graph structure [4, 40, 2]. In recent years, VOS methods have been mainly developed based on deep neural networks (DNN), leading to better results.
+
+Early DNN methods rely on fine-tuning the networks at test time to make segmentation networks focus on a specific object. Among them, OSVOS [7] and MoNet [47] fine-tune pre-trained networks on the first-frame ground-truth at test time. OnAVOS [42] extends the first-frame fine-tuning by introducing an online adaptation mechanism. Following these approaches, MaskTrack [29] and PReM [24] utilize optical flow to help propagate the segmentation mask from one frame to the next. Despite achieving promising results, the test-time fine-tuning restricts the network efficiency.
+
+Recent works aim to achieve a better run-time and avoid using online fine-tuning. OSMN [49] employs one convolutional network to extract object embedding and another one to guide segmentation predictions. PML [9] learns pixel-wise embedding with a nearest neighbor classifier, and VideoMatch [18] uses a soft matching layer that maps the pixels of the current frame to the first frame in a learned embedding space. Following PML and VideoMatch, FEELVOS [41] and CFBI [50, 51] extend the pixel-level matching mechanism by additionally matching between the current frame and the previous frame. RGMP [46] also gathers guidance information from both the first frame and the previous frame but uses a siamese encoder with two shared streams. STM [26] and its following works (e.g., EGMN [23] and KMN [34]) leverage a memory network to embed past-frame predictions into memory and apply a non-local attention mechanism on the memory to decode the segmentation of the current frame. SST [13] utilizes attention mechanisms in a different way, i.e., transformer blocks [39] are used to extract pixel-level affinity maps and spatial-temporal features. The features are target-agnostic, instead of target-aware like our LSTT, since the mask information in past frames is not propagated and aggregated in the blocks. Instead of using matching mechanisms, LWL [6] proposes to use an online few-shot learner to learn to decode object segmentation.
+
+The above methods learn to decode features with a single positive object and thus have to match and segment each target separately under multi-object scenarios, consuming multiple times computing resources of single-object cases. The problem restricts the application and development of the VOS with multiple targets. Hence, we propose our AOT to associate and decode multiple targets uniformly and simultaneously, as efficiently as processing a single object.
+
+Visual Transformers. Transformers [39] was proposed to build hierarchical attention-based networks for machine translation. Similar to Non-local Neural Networks [43], transformer blocks compute correlation with all the input elements and aggregate their information by using attention mechanisms [5]. Compared to RNNs, transformer networks model global correlation or attention in parallel, leading to better memory efficiency, and thus have been widely used in natural language processing (NLP) tasks [11, 32, 37]. Recently, transformer blocks were introduced to many computer vision tasks, such as image classification [12, 38, 22], object detection [8]/segmentation [44], and image generation [27], and have shown promising performance compared to CNN-based networks.
+
+Many VOS methods [19, 26, 23, 34] have utilized attention mechanisms to match the object features and propagate the segmentation mask from past frames to the current frames. Nevertheless, these methods consider only one positive target in the attention processes, and how to build hierarchical attention-based propagation has been rarely studied. In this paper, we carefully design a long shortterm transformer block, which can effectively construct multi-object matching and propagation within hierarchical structures for VOS.
+
+# 3 Revisit Previous Solutions for Video Object Segmentation
+
+In VOS, many common video scenarios have multiple targets or objects required for tracking and segmenting. Benefit from deep networks, current state-of-the-art VOS methods [26, 50] have achieved promising performance. Nevertheless, these methods focus on matching and decoding a single object. Under a multi-object scenario, they thus have to match each object independently and ensemble all the single-object predictions into a multi-object prediction, as demonstrated in Fig. 1a. Let $F ^ { \mathcal { N } }$ denotes a VOS network for predicting single-object segmentation, and $A$ is an ensemble function such as sof tmax or the soft aggregation [26], the formula of such a post-ensemble manner for processing $N$ objects is like,
+
+$$
+Y ^ { \prime } = A ( F ^ { \mathcal { N } } ( I ^ { t } , I ^ { \mathbf { m } } , Y _ { 1 } ^ { \mathbf { m } } ) , . . . , F ^ { \mathcal { N } } ( I ^ { t } , I ^ { \mathbf { m } } , Y _ { N } ^ { \mathbf { m } } ) ) ,
+$$
+
+![](images/81621c58233797a6e0afd78c46fa2910bf9e19ffc51406290c92eaa29bb254e6.jpg)  
+Figure 2: (a) The overview of our Associating Objects with Transformers (AOT). The multi-object masks are embedded by using our Identification mechanism. Moreover, a $L$ -layer Long ShortTerm Transformer is responsible for matching multiple objects uniformly and hierarchically. (b) An illustration of the IDentity assignment (ID) designed for transferring a $N .$ -object mask into an identification embedding. (c) The structure of an LSTT block. LN: layer normalization [3].
+
+where $I ^ { t }$ and $I ^ { \mathbf { m } }$ denote the image of the current frame and memory frames respectively, and $\{ Y _ { 1 } ^ { \mathbf { m } } , . . . , Y _ { N } ^ { \mathbf { m } } \}$ are the memory masks (containing the given reference mask and past predicted masks) of all the $N$ objects. This manner extends networks designed for single-object VOS into multi-object applications, so there is no need to adapt the network for different object numbers.
+
+Although the above post-ensemble manner is prevalent and straightforward in the VOS field, processing multiple objects separately yet in parallel requires multiple times the amount of GPU memory and computation for matching a single object and decoding the segmentation. This problem restricts the training and application of VOS under multi-object scenarios when computing resources are limited. To make the multi-object training and inference as efficient as single-object ones, an expected solution should be capable of associating and decoding multiple objects uniformly instead of individually. To achieve such an objective, we propose an identification mechanism to embed the masks of any number (required to be smaller than a pre-defined large number) of targets into the same high-dimensional space. Based on the identification mechanism, a novel and efficient framework, i.e., Associating Objects with Transformers (AOT), is designed for propagating all the object embeddings uniformly and hierarchically, from memory frames to the current frame.
+
+As shown in Fig. 1b, our AOT associates and segments multiple objects within an end-to-end framework. For the first time, processing multiple objects can be as efficient as processing a single object (Fig. 1c). Compared to previous methods, our training under multi-object scenarios is also more efficient since AOT can associate multiple object regions and learn contrastive feature embeddings among them uniformly.
+
+# 4 Associating Objects with Transformers
+
+In this section, we introduce our identification mechanism proposed for efficient multi-object VOS. Then, we design a new VOS framework, i.e., long short-term transformer, based on the identification mechanism for constructing hierarchical multi-object matching and propagation.
+
+# 4.1 Identification Mechanism for Multi-object Association
+
+Many recent VOS methods [26, 23, 34] utilized attention mechanisms and achieved promising results. To formulate, we define $Q \in \mathbb { R } ^ { H W \times C }$ , $K \in \mathbb { R } ^ { T H W \times C }$ , and $V \in \mathbb { R } ^ { T H W \times C }$ as the query embedding of the current frame, the key embedding of the memory frames, and the value embedding of the memory frames respectively, where $T$ , $H$ , $W$ , $C$ denote the temporal, height, width, and channel dimensions. The formula of a common attention-based matching and propagation is,
+
+$$
+A t t ( Q , K , V ) = C o r r ( Q , K ) V = s o f t m a x ( \frac { Q K ^ { t r } } { \sqrt { C } } ) V ,
+$$
+
+where a matching map is calculated by the correlation function Corr, and then the value embedding, $V$ , will be propagated into each location of the current frame.
+
+In the common single-object propagation [26], the binary mask information in memory frames is embedded into $V$ with an additional memory encoder network and thus can also be propagated to the current frame by using Eq. 2. A convolutional decoder network following the propagated feature will decode the aggregated feature and predict the single-object probability logit of the current frame.
+
+The main problem of propagating and decoding multi-object mask information in an end-to-end network is how to adapt the network to different target numbers. To overcome this problem, we propose an identification mechanism consisting of identification embedding and decoding based on attention mechanisms.
+
+First, an Identification Embedding mechanism is proposed to embed the masks of multiple different targets into the same feature space for propagation. As seen in Fig. 2b, we initialize an identity bank, $D \in \mathbb { R } ^ { M \times C }$ , where $M$ identification vectors with $C$ dimensions are stored. For embedding multiple different target masks, each target will be randomly assigned a different identification vector. Assuming $N$ $N < M )$ targets are in the video scenery, the formula of embedding the targets’ one-hot mask, $\bar { Y } \in \{ 0 , 1 \} ^ { T H W \times N }$ , into a identification embedding, $E \in \mathbb { R } ^ { T H W \times C }$ , by randomly assigning identification vector from the bank $D$ is,
+
+$$
+E = I D ( Y , D ) = Y P D ,
+$$
+
+where $P \in \{ 0 , 1 \} ^ { N \times M }$ is a random permutation matrix, satisfying that $P ^ { t r } P$ is equal to a $M \times M$ unit matrix, for randomly selecting $N$ identification embeddings. After the $I D$ assignment, different target has different identification embedding, and thus we can propagate all the target identification information from memory frames to the current frame by attaching the identification embedding $E$ with the attention value $V$ , i.e.,
+
+$$
+V ^ { \prime } = A t t I D ( Q , K , V , Y | D ) = A t t ( Q , K , V + I D ( Y , D ) ) = A t t ( Q , K , V + E ) ,
+$$
+
+where $V ^ { \prime } \in \mathbb { R } ^ { H W \times C }$ aggregates all the multiple targets’ embeddings from the propagation.
+
+For Identification Decoding, i.e., predicting all the targets’ probabilities from the aggregated feature $V ^ { \prime }$ , we firstly predict the probability logit for every identity in the bank $D$ by employing a convolutional decoding network $\hat { F } ^ { \mathcal { D } }$ , and then select the assigned ones and calculate the probabilities, i.e.,
+
+$$
+Y ^ { \prime } = s o f t m a x ( P F ^ { \mathcal { D } } ( V ^ { \prime } ) ) = s o f t m a x ( P L ^ { D } ) ,
+$$
+
+where $L ^ { D } \in \mathbb { R } ^ { H W \times M }$ is all the $M$ identities’ probability logits, $P$ is the same as the selecting matrix used in the identity assignment (Eq. 3), and $\begin{array} { r } { Y ^ { \dot { \prime } } \in [ 0 , 1 ] ^ { \check { H } W \times N } } \end{array}$ is the probability prediction of all the $N$ targets.
+
+For training, common multi-class segmentation losses, such as cross-entropy loss, can be used to optimize the multi-object $Y ^ { \prime }$ regarding the ground-truth labels. The identity bank $D$ is trainable and randomly initialized at the training beginning. To ensure that all the identification vectors have the same opportunity to compete with each other, we randomly reinitialize the identification selecting matrix $P$ in each video sample and each optimization iteration.
+
+# 4.2 Long Short-Term Transformer for Hierarchical Matching and Propagation
+
+Previous methods [26, 34] always utilize only one layer of attention (Eq. 2) to aggregate singleobject information. In our identification-based multi-object pipeline, we found that a single attention layer cannot fully model multi-object association, which naturally should be more complicated than single-object processes. Thus, we consider constructing hierarchical matching and propagation by using a series of attention layers. Recently, transformer blocks [39] have been demonstrated to be stable and promising in constructing hierarchical attention structures in visual tasks [8, 12]. Based on transformer blocks, we carefully design a Long Short-Term Transformer (LSTT) block for multi-object VOS.
+
+Following the common transformer blocks [39, 11], LSTT firstly employs a self-attention layer, which is responsible for learning the association or correlation among the targets within the current frame. Then, LSTT additionally introduces a long-term attention, for aggregating targets’ information from long-term memory frames and a short-term attention, for learning temporal smoothness from nearby short-term frames. The final module is based on a common 2-layer feed-forward MLP with
+
+GELU [17] non-linearity in between. Fig. 2c shows the structure of an LSTT block. Notably, all these attention modules are implemented in the form of the multi-head attention [39], i.e., multiple attention modules followed by concatenation and a linear projection. Nevertheless, we only introduce their single-head formulas below for the sake of simplicity.
+
+Long-Term Attention is responsible for aggregating targets’ information from past memory frames, which contains the reference frame and stored predicted frames, to the current frame. Since the time intervals between the current frame and past frames are variable and can be long-term, the temporal smoothness is difficult to guarantee. Thus, the long-term attention employs non-local attention like Eq. 2. Let $X _ { l } ^ { t } \in \mathbb { R } ^ { H W \times \mathbf { \breve { C } } }$ denotes the input feature embedding at time $t$ and in block $l$ , where $l \in \{ 1 , . . . , L \}$ is the block index of LSTT, the formula of the long-term attention is,
+
+$$
+A t t L T ( X _ { l } ^ { t } , X _ { l } ^ { \mathbf { m } } , Y ^ { \mathbf { m } } ) = A t t I D ( X _ { l } ^ { t } W _ { l } ^ { K } , X _ { l } ^ { \mathbf { m } } W _ { l } ^ { K } , X _ { l } ^ { \mathbf { m } } W _ { l } ^ { V } , Y ^ { \mathbf { m } } | D ) ,
+$$
+
+where $X _ { l } ^ { \mathbf { m } } = C o n c a t ( X _ { l } ^ { m _ { 1 } } , . . . , X _ { l } ^ { m _ { T } } )$ and $Y ^ { \mathbf { m } } = { C o n c a t } ( Y ^ { m _ { 1 } } , . . . , Y ^ { m _ { T } } )$ are the input feature embeddings and target masks of memory frames with indices $\mathbf { m } = \{ m _ { 1 } , . . . , m _ { T } \}$ . Besides, $W _ { l } ^ { K } \in$ $\mathbb { R } ^ { C \times C _ { k } }$ and $W _ { l } ^ { V } \in \mathbb { R } ^ { C \times C _ { v } }$ are trainable parameters of the space projections for matching and propagation, respectively. Instead of using different projections for ${ \bar { X } } _ { l } ^ { t }$ and $X _ { l } ^ { \mathbf { m } }$ , we found the training of LSTT is more stable with a siamese-like matching, $i . e .$ , matching between the features within the same embedding space $\mathit { l }$ -th features with the same projection of $\breve { W } _ { l } ^ { K }$ ).
+
+Short-Term Attention is employed for aggregating information in a spatial-temporal neighbourhood for each current-frame location. Intuitively, the image changes across several contiguous video frames are always smooth and continuous. Thus, the target matching and propagation in contiguous frames can be restricted in a small spatial-temporal neighborhood, leading to better efficiency than non-local processes. Considering $n$ neighbouring frames with indices $\mathbf { n } = \{ t - 1 , . . . , t - n \}$ are in the spatialtemporal neighbourhood, the features and masks of these frames are $X _ { l } ^ { \mathbf { n } } = C o n c a t ( X _ { l } ^ { t - 1 } , . . . , X _ { l } ^ { t - n } )$ and $Y ^ { \mathbf { n } } = C o n c a t ( Y ^ { t - 1 } , . . . , Y ^ { t - n } )$ , and then the formula of the short-term attention at each spatial location $p$ is,
+
+$$
+A t t S T ( X _ { l } ^ { t } , X _ { l } ^ { { \bf n } } , Y ^ { { \bf n } } | p ) = A t t L T ( X _ { l , p } ^ { t } , X _ { l , N ( p ) } ^ { { \bf n } } , Y _ { l , N ( p ) } ^ { { \bf n } } ) ,
+$$
+
+where $X _ { l , p } ^ { t } \in \mathbb { R } ^ { 1 \times C }$ is the feature of $X _ { l } ^ { t }$ at location $p , { \mathcal { N } } ( p )$ is a $\lambda \times \lambda$ spatial neighbourhood l,p centered at location $p$ , and thus $X _ { l , N ( p ) } ^ { \mathbf { n } }$ land $Y _ { l , \mathcal { N } ( p ) } ^ { \mathbf { n } }$ are the features and masks of the spatial-temporal neighbourhood, respectively, with a shape of $n \lambda ^ { 2 } \times C$ or $n \lambda ^ { 2 } \times N$ .
+
+When extracting features of the first frame $t = 1$ , there is no memory frames or previous frames, and hence we use $\bar { X } _ { l } ^ { 1 }$ to replace $X _ { l } ^ { \mathbf { m } }$ and $X _ { l } ^ { \mathbf { n } }$ . In other words, the long-term attention and the short-term attention are changed into self-attentions without adjusting the network structures and parameters.
+
+# 5 Implementation Details
+
+Network Details: For sufficiently validating the effectiveness of our identification mechanism and LSTT, we mainly use light-weight backbone encoder, MobileNet-V2 [33], and decoder, FPN [20] with Group Normalization [45]. The spatial neighborhood size $\lambda$ is set to 15, and the number of identification vectors, $M$ , is set to 10, which is consistent with the maximum object number in the benchmarks [48, 31]. AOT performs well with PaddlePaddle [1] and PyTorch [28]. More details can be found in the supplementary material.
+
+Architecture Variants: We build several AOT variant networks with different LSTT layer number $L$ or long-term memory size m. The hyper-parameters of these variants are: (1) AOT-Tiny: $L = 1$ , $\mathbf { m } = \bar { \{ 1 \} }$ ; (2) AOT-Small: $L = 2$ , $\mathbf { m } = \{ 1 \}$ ; (3) AOT-Base: $L = 3$ , $\mathbf { m } = \{ 1 \}$ ; (4) AOT-Large: $L = 3$ , $\begin{array} { r } { \dot { \mathbf { m } } = \{ 1 , 1 + \delta , 1 + 2 \delta , 1 + 3 \delta , . . . \} . } \end{array}$ . In the experiments, we also equip AOT-L with ResNet50 (R50) [16] or Swin-B [22].
+
+AOT-S is a small model with only 2 layers of LSTT block. Compared to AOT-S, AOT-T utilizes only 1 layer of LSTT, and AOT- $\mathbf { B } / \mathbf { L }$ uses 3 layers. In AOT-T/S/B, only the first frame is considered into long-term memory, which is similar to [41, 50], leading to a smooth efficiency. In AOT-L, the predicted frames are stored into long-term memory per $\delta$ frames, following the memory reading strategy [26]. We set $\delta$ to 2/5 for training/testing.
+
+Training Details: Following [46, 26, 23, 34], the training stage is divided into two phases: (1) pre-training on sythetic video sequence generated from static image datasets [14, 21, 10, 36, 15] by randomly applying multiple image augmentations [46]. (2) main training on the VOS benchmarks [48, 31] by randomly applying video augmentations [50]. More details are in the supplementary material.
+
+Table 1: The quantitative evaluation on multi-object benchmarks, YouTube-VOS [48] and DAVIS 2017 [31]. Y: using YouTube-VOS for training. ∗: using 600p instead of 480p videos in inference. $^ \ddag$ : timing extrapolated from single-object speed assuming linear scaling in the number of objects.   
+
+<table><tr><td colspan="7">(a) YouTube-VOS</td><td colspan="5">(b) DAVIS 2017</td></tr><tr><td></td><td></td><td colspan="2">Seen</td><td colspan="3">Unseen</td><td>Methods</td><td>J&amp;F</td><td>J</td><td>F</td><td>FPS</td></tr><tr><td>Methods</td><td>J&amp;F</td><td>J</td><td>F</td><td>J</td><td>F</td><td>FPS</td><td colspan="3">Validation2017 Split</td><td></td><td></td></tr><tr><td colspan="3">Validation 2018 Split</td><td></td><td></td><td></td><td></td><td>CFBI [50] (Y)</td><td>81.9</td><td>79.3</td><td>84.5</td><td>5.9</td></tr><tr><td>STM[ICCV19] [26]</td><td>79.4</td><td>79.7</td><td>84.2</td><td>72.8</td><td>80.9</td><td>-</td><td>SST[13](Y)</td><td>82.5</td><td>79.9</td><td>85.1</td><td>1</td></tr><tr><td>KMN[ECCV20] [34]</td><td>81.4</td><td>81.4</td><td>85.6</td><td>75.3</td><td>83.3</td><td></td><td>KMN [34]</td><td>76.0</td><td>74.2</td><td>77.8</td><td>4.2t</td></tr><tr><td>CFBI[ECCV20] [50]</td><td>81.4</td><td>81.1</td><td>85.8</td><td>75.3</td><td>83.4</td><td>3.4</td><td>KMN [34] (Y)</td><td>82.8</td><td>80.0</td><td>85.6</td><td>4.2</td></tr><tr><td>LWL[ECCV20] [6]</td><td>81.5</td><td>80.4</td><td>84.9</td><td>76.4</td><td>84.4</td><td>1</td><td>CFBI+ [51] (Y)</td><td>82.9</td><td>80.1</td><td>85.7</td><td>5.6</td></tr><tr><td>SST[CVPR21] [13]</td><td>81.7</td><td>81.2</td><td>1</td><td>76.0</td><td>1</td><td>1</td><td>AOT-T (Y)</td><td>79.9</td><td>77.4</td><td>82.3</td><td>51.4</td></tr><tr><td>CFBI+[TPAMI21] [51]</td><td>82.8</td><td>81.8</td><td>86.6</td><td>77.1</td><td>85.6</td><td>4.0</td><td>AOT-S</td><td>79.2</td><td>76.4</td><td>82.0</td><td>40.0</td></tr><tr><td>AOT-T</td><td>80.2</td><td>80.1</td><td>84.5</td><td>74.0</td><td>82.2</td><td>41.0</td><td>AOT-S (Y)</td><td>81.3</td><td>78.7</td><td>83.9</td><td>40.0</td></tr><tr><td>AOT-S</td><td>82.6</td><td>82.0</td><td>86.7</td><td>76.6</td><td>85.0</td><td>27.1</td><td>AOT-B (Y)</td><td>82.5</td><td>79.7</td><td>85.2</td><td>29.6</td></tr><tr><td>AOT-B</td><td>83.5</td><td>82.6</td><td>87.5</td><td>77.7</td><td>86.0</td><td>20.5</td><td>AOT-L (Y)</td><td>83.8</td><td>81.1</td><td>86.4</td><td>18.7</td></tr><tr><td>AOT-L</td><td>83.8</td><td>82.9</td><td>87.9</td><td>77.7</td><td>86.5</td><td>16.0</td><td>R50-AOT-L (Y)</td><td>84.9</td><td>82.3</td><td>87.5</td><td>18.0</td></tr><tr><td>R50-AOT-L</td><td>84.1</td><td>83.7</td><td>88.5</td><td>78.1</td><td>86.1</td><td>14.9</td><td>SwinB-AOT-L (Y)</td><td>85.4</td><td>82.4</td><td>88.4</td><td>12.1</td></tr><tr><td>SwinB-AOT-L</td><td>84.5</td><td>84.3</td><td>89.3</td><td>77.9</td><td>86.4</td><td>9.3</td><td>Testing 2017 Split</td><td></td><td></td><td></td><td></td></tr><tr><td colspan="5">Validation 2019 Split</td><td></td><td colspan="5">71.4 78.7</td></tr><tr><td>CFBI[ECCV20] [50]</td><td>81.0</td><td>80.6</td><td>85.1</td><td>75.2</td><td>83.0</td><td>3.4</td><td>CFBI [50] (Y) CFBI* [50](Y)</td><td>75.0 76.6</td><td>73.0</td><td>80.1</td><td>5.3 2.9</td></tr><tr><td>SST[CVPR21] [13]</td><td>81.8</td><td>80.9</td><td>1</td><td>76.6</td><td>-</td><td>-</td><td>KMN* [34] (Y)</td><td>77.2</td><td>74.1</td><td>80.3</td><td>1</td></tr><tr><td>CFBI+[TPAMI21] [51]</td><td>82.6</td><td>81.7</td><td>86.2</td><td>77.1</td><td>85.2</td><td>4.0</td><td>CFBI+* [51](Y)</td><td>78.0</td><td>74.4</td><td>81.6</td><td>3.4</td></tr><tr><td>AOT-T</td><td>79.7</td><td>79.6</td><td>83.8</td><td>73.7</td><td>81.8</td><td>41.0</td><td>AOT-T (Y)</td><td>72.0</td><td>68.3</td><td>75.7</td><td>51.4</td></tr><tr><td>AOT-S</td><td>82.2</td><td>81.3</td><td>85.9</td><td>76.6</td><td>84.9</td><td>27.1</td><td>AOT-S (Y)</td><td>73.9</td><td>70.3</td><td>77.5</td><td>40.0</td></tr><tr><td>AOT-B</td><td>83.3</td><td>82.4</td><td>87.1</td><td>77.8</td><td>86.0</td><td>20.5</td><td>AOT-B (Y)</td><td>75.5</td><td>71.6</td><td>79.3</td><td>29.6</td></tr><tr><td>AOT-L</td><td>83.7</td><td>82.8</td><td>87.5</td><td>78.0</td><td>86.7</td><td>16.0</td><td>AOT-L (Y)</td><td>78.3</td><td>74.3</td><td>82.3</td><td>18.7</td></tr><tr><td>R50-AOT-L</td><td>84.1</td><td>83.5</td><td>88.1</td><td>78.4</td><td>86.3</td><td>14.9</td><td>R50-AOT-L (Y)</td><td>79.6</td><td>75.9</td><td>83.3</td><td>18.0</td></tr><tr><td>SwinB-AOT-L</td><td>84.5</td><td>84.0</td><td>88.8</td><td>78.4</td><td>86.7</td><td>9.3</td><td>SwinB-AOT-L (Y)</td><td>81.2</td><td>77.3</td><td>85.1</td><td>12.1</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+# 6 Experimental Results
+
+We evaluate AOT on popular multi-object benchmarks, YouTube-VOS [48] and DAVIS 2017 [31], and single-object benchmark, DAVIS 2016 [30]. For YouTube-VOS experiments, we train our models on the YouTube-VOS 2019 training split. For DAVIS, we train on the DAVIS-2017 training split. When evaluating YouTube-VOS, we use the default 6FPS videos, and all the videos are restricted to be smaller than $1 . 3 \times 4 8 0 p$ resolution. As to DAVIS, the default 480p 24FPS videos are used.
+
+The evaluation metric is the $\mathcal { I }$ score, calculated as the average Intersect over Union (IoU) score between the prediction and the ground truth mask, and the $\mathcal { F }$ score, calculated as an average boundary similarity measure between the boundary of the prediction and the ground truth, and their mean value, denoted as $\mathcal { I } \& \mathcal { F }$ . We evaluate all the results on official evaluation servers or with official tools.
+
+# 6.1 Compare with the State-of-the-art Methods
+
+YouTube-VOS [48] is the latest large-scale benchmark for multi-object video segmentation and is about 37 times larger than DAVIS 2017 (120 videos). Specifically, YouTube-VOS contains 3471 videos in the training split with 65 categories and 474/507 videos in the validation 2018/2019 split with additional 26 unseen categories. The unseen categories do not exist in the training split to evaluate algorithms’ generalization ability.
+
+As shown in Table 1a, AOT variants achieve superior performance on YouTube-VOS compared to the previous state-of-the-art methods. With our identification mechanism, AOT-S ( $8 2 . 6 \%$ J &F )
+
+![](images/d28db8d7b0e6eb707adaa9bbbec03cc11f76dfc9cca42cb535353a8852c56975.jpg)  
+Figure 3: Qualitative results. (top) Compared with CFBI [50], AOT performs better when segmenting multiple highly similar objects (carousels and zebras). (bottom) AOT fails to segment some tiny objects (ski poles and watch) since AOT has no specific design for processing rare tiny objects.
+
+is comparable with $\mathrm { C F B I + }$ [51] $( 8 2 . 8 \% )$ while running about $7 \times$ faster (27.1 vs 4.0FPS). By using more LSTT blocks, AOT-B improves the performance to $8 3 . 5 \%$ . Moreover, AOT-L further improves both the seen and unseen scores by utilizing the memory reading strategy, and our R50-AOT-L $( 8 4 . 1 \% / 8 4 . 1 \% )$ significantly outperforms the previous methods on the validation 2018/2019 split while maintaining an efficient speed (14.9FPS).
+
+DAVIS 2017 [31] is a multi-object extension of DAVIS 2016. The validation split of DAVIS 2017 consists of 30 videos with 59 objects, and the training split contains 60 videos with 138 objects. Moreover, the testing split contains 30 more challenging videos with 89 objects in total.
+
+Table 1b shows that our R50-AOT-L $( \mathbf { Y } )$ surpasses all the competitors on both the DAVIS-2017 validation $( \mathbf { 8 4 . 9 \% } )$ and testing $( 7 9 . 6 \% )$ splits and maintains an efficient speed (18.0FPS). Notably, such a multiobject speed is the same as our single-object speed on DAVIS 2016. For the first time, processing multiple objects can be as efficient as processing a single object over the AOT framework. We also evaluate our method without training with YouTube-VOS, and AOT-S $( 7 9 . 2 \% )$ performs much better than KMN [34] $( 7 6 . 0 \% )$ by $+ 3 . 2 \%$ .
+
+Table 2: The quantitative evaluation on the single-object DAVIS 2016 [30].   
+
+<table><tr><td>Methods</td><td>J&amp;F</td><td>J</td><td>F</td><td>FPS</td></tr><tr><td>CFBI+ [51] (Y)</td><td>89.9 90.5</td><td>88.7 89.5</td><td>91.1 91.5</td><td>5.9 8.3</td></tr><tr><td>KMN [34](Y) AOT-T (Y)</td><td>86.8</td><td>86.1</td><td>87.4</td><td>51.4</td></tr><tr><td>AOT-S (Y)</td><td>89.4</td><td></td><td>88.6 90.2</td><td>40.0</td></tr><tr><td>AOT-B (Y)</td><td>89.9</td><td></td><td>88.7 91.1</td><td>29.6</td></tr><tr><td>AOT-L (Y)</td><td>90.4</td><td>89.6 91.1</td><td></td><td>18.7</td></tr><tr><td>R50-AOT-L (Y) SwinB-AOT-L (Y)</td><td>91.1 92.0</td><td></td><td>90.1 92.11 90.7 93.3 12.1</td><td>18.0</td></tr></table>
+
+DAVIS 2016 [30] is a single-object benchmark containing 20 videos in the validation split. Although our AOT aims at improving multi-object video segmentation, we also achieve a new state-of-the-art performance on DAVIS 2016 (R50-AOT-L $( \mathbf { Y } )$ , $91 . 1 \%$ ). Under single-object scenarios, the multiobject superiority of AOT is limited, but R50-AOT-L still maintains an about $2 \times$ efficiency compared to KMN $( 1 8 . 0 \nu s 8 . 3 \mathrm { F P S } )$ ). Furthermore, our smaller variant, AOT-B $( 8 9 . 9 \% )$ , achieves comparable performance with CFBI+ $( 8 9 . 9 \% )$ while running $5 \times$ faster (29.6 vs 5.9FPS).
+
+Apart from the above results, replacing the AOT encoder from commonly used ResNet50 to SwinB can further boost our performance to higher level (Table 1a, 1b, and 2).
+
+Qualitative results: Fig. 3 visualizes some qualitative results in comparison with CFBI [50], which only associates each object with its relative background. As demonstrated, CFBI is easier to confuse multiple highly similar objects. In contrast, our AOT tracks and segments all the targets accurately by associating all the objects uniformly. However, AOT fails to segment some tiny objects (ski poles and watch) since we do not make special designs for tiny objects.
+
+# 6.2 Ablation Study
+
+In this section, we analyze the main components and hyper-parameters of AOT and evaluate their impact on the VOS performance in Table 3.
+
+Identity number: The number of the identification vectors, $M$ , have to be larger than the object number in videos. Thus, we set $M$ to 10 in default to be consistent with the maximum object number
+
+(a) Identity number
+
+Table 3: Ablation study. The experiments are based on AOT-S and conducted on the validation 2018 split of YouTube-VOS [48] without pre-training on synthetic videos. Self: the position embedding type used in the self-attention. Rel: use relative positional embedding [35] on the local attention.   
+
+<table><tr><td>M J&amp;F</td><td>Jseen Junseen</td></tr><tr><td>10 80.3</td><td>80.6</td></tr><tr><td>79.0</td><td>73.7 79.4</td></tr><tr><td></td><td>72.1 70.8</td></tr><tr><td>20 78.3 30 77.2</td><td>79.4 78.5 70.2</td></tr></table>
+
+(c) Local frame number   
+
+<table><tr><td>n J&amp;F</td><td>Jseen</td><td>Junseen</td></tr><tr><td>1 80.3</td><td>80.6</td><td>73.7</td></tr><tr><td>2</td><td>80.0 79.8</td><td>73.7</td></tr><tr><td>3 79.1</td><td>80.0</td><td>72.2</td></tr><tr><td>0 74.3</td><td>74.9</td><td>67.6</td></tr></table>
+
+(b) Local window size   
+(d) LSTT block number   
+
+<table><tr><td>入 J&amp;F</td><td>Jseen</td><td>Junseen</td></tr><tr><td>15 80.3</td><td>80.6</td><td>73.7</td></tr><tr><td>11</td><td>78.8 79.5</td><td>71.9</td></tr><tr><td>7 78.3</td><td>79.3</td><td>70.9</td></tr><tr><td>0 74.3</td><td>74.9</td><td>67.6</td></tr></table>
+
+(e) Positional embedding   
+
+<table><tr><td colspan="3"> Self Rel J&amp;F gseen</td><td></td><td>Junseen</td></tr><tr><td>sine</td><td>√</td><td>80.3</td><td>80.6</td><td>73.7</td></tr><tr><td>none</td><td>√</td><td>80.1</td><td>80.4</td><td>73.5</td></tr><tr><td>sine</td><td>1</td><td>79.7</td><td>80.1</td><td>72.9</td></tr></table>
+
+<table><tr><td>L J&amp;F Jseen</td><td></td><td>Junseen</td><td>FPS Param</td></tr><tr><td>2</td><td>80.3 80.6</td><td>73.7</td><td>27.17.0M</td></tr><tr><td>3</td><td>80.9</td><td>81.1 74.0</td><td>20.58.3M</td></tr><tr><td>1</td><td>77.9 78.8</td><td>71.0</td><td>41.0 5.7M</td></tr></table>
+
+in the benchmarks [48, 31]. As seen in Table 3a, $M$ larger than 10 leads to worse performance since (1) no training video contains so many objects; (2) embedding more than 10 objects into the space with only 256 dimensions is difficult.
+
+Local window size: Table 3b shows that larger local window size, $\lambda$ , results in better performance. Without the local attention, $\lambda = 0$ , the performance of AOT significantly drops from $8 0 . 3 \%$ to $7 4 . 3 \%$ , which demonstrates the necessity of the local attention.
+
+Local frame number: In Table 3c, we also try to employ more previous frames in the local attention, but using only the $t - 1$ frame $( 8 0 . 3 \% )$ performs better than using 2/3 frames $( 8 0 . 0 \% / 7 9 . 1 \% )$ . A possible reason is that the longer the temporal interval, the more intense the motion between frames, so it is easier to introduce more errors in the local matching when using an earlier previous frame.
+
+LSTT block number: As shown in Table 3d, the AOT performance increases by using more LSTT blocks. Notably, the AOT with only one LSTT block $( 7 7 . 9 \% )$ reaches a fast real-time speed (41.0FPS) on YouTube-VOS, although the performance is $- 2 . 4 \%$ worse than AOT-S $( 8 0 . 3 \% )$ . By adjusting the LSTT block number, we can flexibly balance the accuracy and speed of AOT.
+
+Position embedding: In our default setting, we apply fixed sine spatial positional embedding to the self-attention following [8], and our local attention is equipped with learned relative positional embedding [35]. The ablation study is shown in Table 3e, where removing the sine embedding decreases the performance to $8 0 . 1 \%$ slightly. In contrast, the relative embedding is more important than the sine embedding. Without the relative embedding, the performance drops to $7 9 . 7 \%$ , which means the motion relationship between adjacent frames is helpful for local attention. We also tried to apply learned positional embedding to self-attention modules, but no positive effect was observed.
+
+# 7 Conclusion
+
+This paper proposes a novel and efficient approach for video object segmentation by associating objects with transformers and achieves superior performance on three popular benchmarks. A simple yet effective identification mechanism is proposed to associate, match, and decode all the objects uniformly under multi-object scenarios. For the first time, processing multiple objects in VOS can be efficient as processing a single object by using the identification mechanism. In addition, a long short-term transformer is designed for constructing hierarchical object matching and propagation for VOS. The hierarchical structure allows us to flexibly balance AOT between real-time speed and stateof-the-art performance by adjusting the LSTT number. We hope the identification mechanism will help ease the future study of multi-object VOS and related tasks (e.g., video instance segmentation, interactive VOS, and multi-object tracking), and AOT will serve as a solid baseline.
+
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parse/train/hl3v8io3ZYt/hl3v8io3ZYt_content_list.json

@@ -0,0 +1,1176 @@
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+    {
+        "type": "text",
+        "text": "Associating Objects with Transformers for Video Object Segmentation ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Zongxin Yang1,2, Yunchao Wei3,4, Yi Yang1 ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 CCAI, College of Computer Science and Technology, Zhejiang University 2 Baidu Research 3 Institute of Information Science, Beijing Jiaotong University 4 Beijing Key Laboratory of Advanced Information Science and Network {zongxinyang1996, wychao1987, yee.i.yang}@gmail.com ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            347,
+            535,
+            363
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This paper investigates how to realize better and more efficient embedding learning to tackle the semi-supervised video object segmentation under challenging multi-object scenarios. The state-of-the-art methods learn to decode features with a single positive object and thus have to match and segment each target separately under multi-object scenarios, consuming multiple times computing resources. To solve the problem, we propose an Associating Objects with Transformers (AOT) approach to match and decode multiple objects uniformly. In detail, AOT employs an identification mechanism to associate multiple targets into the same high-dimensional embedding space. Thus, we can simultaneously process multiple objects’ matching and segmentation decoding as efficiently as processing a single object. For sufficiently modeling multi-object association, a Long Short-Term Transformer is designed for constructing hierarchical matching and propagation. We conduct extensive experiments on both multi-object and single-object benchmarks to examine AOT variant networks with different complexities. Particularly, our R50-AOT-L outperforms all the state-of-the-art competitors on three popular benchmarks, i.e., YouTube-VOS $( 8 4 . 1 \\% ~ \\mathcal { I } \\& \\mathcal { F } )$ , DAVIS 2017 $( 8 4 . 9 \\% )$ , and DAVIS 2016 $( 9 1 . 1 \\% )$ , while keeping more than $3 \\times$ faster multi-object run-time. Meanwhile, our AOT-T can maintain real-time multi-object speed on the above benchmarks. Based on AOT, we ranked $\\mathbf { 1 ^ { s t } }$ in the 3rd Large-scale VOS Challenge. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Video Object Segmentation (VOS) is a fundamental task in video understanding with many potential applications, including augmented reality [25] and self-driving cars [52]. The goal of semi-supervised VOS, the main task in this paper, is to track and segment object(s) across an entire video sequence based on the object mask(s) given at the first frame. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Thanks to the recent advance of deep neural networks, many deep learning based VOS algorithms have been proposed recently and achieved promising performance. STM [26] and its following works [34, 23] leverage a memory network to store and read the target features of predicted past frames and apply a non-local attention mechanism to match the target in the current frame. FEELVOS [41] and CFBI [50, 51] utilize global and local matching mechanisms to match target pixels or patches from both the first and the previous frames to the current frame. ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Even though the above methods have achieved significant progress, the above methods learn to decode scene features that contain a single positive object. Thus under a multi-object scenario, they have to match each object independently and ensemble all the single-object predictions into a multi-object segmentation, as shown in Fig. 1a. Such a post-ensemble manner eases network architectures’ design since the networks are not required to adapt the parameters or structures for different object numbers. However, modeling multiple objects independently, instead of uniformly, is inefficient in exploring multi-object contextual information to learn a more robust feature representation for VOS. In addition, processing multiple objects separately yet in parallel requires multiple times the amount of GPU memory and computation for processing a single object. This problem restricts the training and application of VOS under multi-object scenarios, especially when computing resources are limited. ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "image",
+        "img_path": "images/4b20923d50ad97a87da9b6849c8352dfcb5e2697e5936a15d3f25e62b6e5b068.jpg",
+        "image_caption": [
+            "Figure 1: VOS methods (e.g., [50, 34]) process multi-object scenarios in a post-ensemble manner (a). In contrast, our AOT associates all the objects uniformly (b), leading to better efficiency (c). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To solve the problem, Fig. 1b demonstrates a feasible approach to associate and decode multiple objects uniformly in an end-to-end framework. Hence, we propose an Associating Objects with Transformers (AOT) approach to match and decode multiple targets uniformly. First, an identification mechanism is proposed to assign each target a unique identity and embed multiple targets into the same feature space. Hence, the network can learn the association or correlation among all the targets. Moreover, the multi-object segmentation can be directly decoded by utilizing assigned identity information. Second, a Long Short-Term Transformer (LSTT) is designed for constructing hierarchical object matching and propagation. Each LSTT block utilizes a long-term attention for matching with the first frame’s embedding and a short-term attention for matching with several nearby frames’ embeddings. Compared to the methods [26, 34] utilizing only one attention layer, we found hierarchical attention structures are more effective in associating multiple objects. ",
+        "bbox": [
+            174,
+            375,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We conduct extensive experiments on two popular multi-object benchmarks for VOS, i.e., YouTubeVOS [48] and DAVIS 2017 [31], to validate the effectiveness and efficiency of the proposed AOT. Even using the light-weight Mobilenet-V2 [33] as the backbone encoder, the AOT variant networks achieve superior performance on the validation 2018 & 2019 splits of the large-scale YouTube-VOS (ours, $\\mathcal { T } \\& \\mathcal { F } 8 2 . 6 \\sim 8 4 . 5 \\%$ & $8 2 . 2 \\sim 8 4 . 5 \\%$ while keeping more than $\\mathbf { 2 \\times }$ faster multi-object run-time $( \\mathbf { 2 7 . 1 } \\sim \\mathbf { 9 . 3 F P S }$ ) compared to the state-of-the-art competitors (e.g., CFBI [50], $8 1 . 4 \\%$ & $8 1 . 0 \\%$ , 3.4FPS). We also achieve new state-of-the-art performance on both the DAVIS-2017 validation $( 8 5 . 4 \\% )$ and testing $( 8 1 . 2 \\% )$ splits. Moreover, AOT is effective under single-object scenarios as well and outperforms previous methods on DAVIS 2016 [30] $( 9 2 . 0 \\% )$ , a popular single-object benchmark. Besides, our smallest variant, AOT-T, can maintain real-time multi-object speed on all above benchmarks (51.4FPS on $4 8 0 \\mathrm { p }$ videos). Particularly, AOT ranked $\\mathbf { 1 ^ { s t } }$ in the Track 1 (Video Object Segmentation) of the 3rd Large-scale Video Object Segmentation Challenge. ",
+        "bbox": [
+            173,
+            534,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Overall, our contributions are summarized as follows: ",
+        "bbox": [
+            176,
+            705,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We propose an identification mechanism to associate and decode multiple targets uniformly for VOS. For the first time, multi-object training and inference can be efficient as single-object ones, as demonstrated in Fig. 1c. • Based on the identification mechanism, we design a new efficient VOS framework, i.e., Long ShortTerm Transformer (LSTT), for constructing hierarchical multi-object matching and propagation. LSTT achieves superior performance on VOS benchmarks [48, 31, 30] while maintaining better efficiency than previous state-of-the-art methods. To the best of our knowledge, LSTT is the first hierarchical framework for object matching and propagation by applying transformers [39] to VOS. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            852,
+            321,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Semi-supervised Video Object Segmentation. Given one or more annotated frames (the first frame in general), semi-supervised VOS methods propagate the manual labeling to the entire video sequence. Traditional methods often solve an optimization problem with an energy defined over a graph structure [4, 40, 2]. In recent years, VOS methods have been mainly developed based on deep neural networks (DNN), leading to better results. ",
+        "bbox": [
+            174,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Early DNN methods rely on fine-tuning the networks at test time to make segmentation networks focus on a specific object. Among them, OSVOS [7] and MoNet [47] fine-tune pre-trained networks on the first-frame ground-truth at test time. OnAVOS [42] extends the first-frame fine-tuning by introducing an online adaptation mechanism. Following these approaches, MaskTrack [29] and PReM [24] utilize optical flow to help propagate the segmentation mask from one frame to the next. Despite achieving promising results, the test-time fine-tuning restricts the network efficiency. ",
+        "bbox": [
+            174,
+            140,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Recent works aim to achieve a better run-time and avoid using online fine-tuning. OSMN [49] employs one convolutional network to extract object embedding and another one to guide segmentation predictions. PML [9] learns pixel-wise embedding with a nearest neighbor classifier, and VideoMatch [18] uses a soft matching layer that maps the pixels of the current frame to the first frame in a learned embedding space. Following PML and VideoMatch, FEELVOS [41] and CFBI [50, 51] extend the pixel-level matching mechanism by additionally matching between the current frame and the previous frame. RGMP [46] also gathers guidance information from both the first frame and the previous frame but uses a siamese encoder with two shared streams. STM [26] and its following works (e.g., EGMN [23] and KMN [34]) leverage a memory network to embed past-frame predictions into memory and apply a non-local attention mechanism on the memory to decode the segmentation of the current frame. SST [13] utilizes attention mechanisms in a different way, i.e., transformer blocks [39] are used to extract pixel-level affinity maps and spatial-temporal features. The features are target-agnostic, instead of target-aware like our LSTT, since the mask information in past frames is not propagated and aggregated in the blocks. Instead of using matching mechanisms, LWL [6] proposes to use an online few-shot learner to learn to decode object segmentation. ",
+        "bbox": [
+            174,
+            229,
+            825,
+            435
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The above methods learn to decode features with a single positive object and thus have to match and segment each target separately under multi-object scenarios, consuming multiple times computing resources of single-object cases. The problem restricts the application and development of the VOS with multiple targets. Hence, we propose our AOT to associate and decode multiple targets uniformly and simultaneously, as efficiently as processing a single object. ",
+        "bbox": [
+            174,
+            443,
+            825,
+            512
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Visual Transformers. Transformers [39] was proposed to build hierarchical attention-based networks for machine translation. Similar to Non-local Neural Networks [43], transformer blocks compute correlation with all the input elements and aggregate their information by using attention mechanisms [5]. Compared to RNNs, transformer networks model global correlation or attention in parallel, leading to better memory efficiency, and thus have been widely used in natural language processing (NLP) tasks [11, 32, 37]. Recently, transformer blocks were introduced to many computer vision tasks, such as image classification [12, 38, 22], object detection [8]/segmentation [44], and image generation [27], and have shown promising performance compared to CNN-based networks. ",
+        "bbox": [
+            173,
+            518,
+            825,
+            630
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Many VOS methods [19, 26, 23, 34] have utilized attention mechanisms to match the object features and propagate the segmentation mask from past frames to the current frames. Nevertheless, these methods consider only one positive target in the attention processes, and how to build hierarchical attention-based propagation has been rarely studied. In this paper, we carefully design a long shortterm transformer block, which can effectively construct multi-object matching and propagation within hierarchical structures for VOS. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Revisit Previous Solutions for Video Object Segmentation ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In VOS, many common video scenarios have multiple targets or objects required for tracking and segmenting. Benefit from deep networks, current state-of-the-art VOS methods [26, 50] have achieved promising performance. Nevertheless, these methods focus on matching and decoding a single object. Under a multi-object scenario, they thus have to match each object independently and ensemble all the single-object predictions into a multi-object prediction, as demonstrated in Fig. 1a. Let $F ^ { \\mathcal { N } }$ denotes a VOS network for predicting single-object segmentation, and $A$ is an ensemble function such as sof tmax or the soft aggregation [26], the formula of such a post-ensemble manner for processing $N$ objects is like, ",
+        "bbox": [
+            173,
+            777,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fe01cde9aa527d55dbb4d443cc8f8d6a99c8296f17c01dfbb964b27d6508fd7e.jpg",
+        "text": "$$\nY ^ { \\prime } = A ( F ^ { \\mathcal { N } } ( I ^ { t } , I ^ { \\mathbf { m } } , Y _ { 1 } ^ { \\mathbf { m } } ) , . . . , F ^ { \\mathcal { N } } ( I ^ { t } , I ^ { \\mathbf { m } } , Y _ { N } ^ { \\mathbf { m } } ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            333,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/81621c58233797a6e0afd78c46fa2910bf9e19ffc51406290c92eaa29bb254e6.jpg",
+        "image_caption": [
+            "Figure 2: (a) The overview of our Associating Objects with Transformers (AOT). The multi-object masks are embedded by using our Identification mechanism. Moreover, a $L$ -layer Long ShortTerm Transformer is responsible for matching multiple objects uniformly and hierarchically. (b) An illustration of the IDentity assignment (ID) designed for transferring a $N .$ -object mask into an identification embedding. (c) The structure of an LSTT block. LN: layer normalization [3]. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            90,
+            823,
+            277
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $I ^ { t }$ and $I ^ { \\mathbf { m } }$ denote the image of the current frame and memory frames respectively, and $\\{ Y _ { 1 } ^ { \\mathbf { m } } , . . . , Y _ { N } ^ { \\mathbf { m } } \\}$ are the memory masks (containing the given reference mask and past predicted masks) of all the $N$ objects. This manner extends networks designed for single-object VOS into multi-object applications, so there is no need to adapt the network for different object numbers. ",
+        "bbox": [
+            174,
+            385,
+            825,
+            440
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Although the above post-ensemble manner is prevalent and straightforward in the VOS field, processing multiple objects separately yet in parallel requires multiple times the amount of GPU memory and computation for matching a single object and decoding the segmentation. This problem restricts the training and application of VOS under multi-object scenarios when computing resources are limited. To make the multi-object training and inference as efficient as single-object ones, an expected solution should be capable of associating and decoding multiple objects uniformly instead of individually. To achieve such an objective, we propose an identification mechanism to embed the masks of any number (required to be smaller than a pre-defined large number) of targets into the same high-dimensional space. Based on the identification mechanism, a novel and efficient framework, i.e., Associating Objects with Transformers (AOT), is designed for propagating all the object embeddings uniformly and hierarchically, from memory frames to the current frame. ",
+        "bbox": [
+            173,
+            446,
+            825,
+            599
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As shown in Fig. 1b, our AOT associates and segments multiple objects within an end-to-end framework. For the first time, processing multiple objects can be as efficient as processing a single object (Fig. 1c). Compared to previous methods, our training under multi-object scenarios is also more efficient since AOT can associate multiple object regions and learn contrastive feature embeddings among them uniformly. ",
+        "bbox": [
+            174,
+            606,
+            825,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 Associating Objects with Transformers ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            694,
+            531,
+            712
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section, we introduce our identification mechanism proposed for efficient multi-object VOS. Then, we design a new VOS framework, i.e., long short-term transformer, based on the identification mechanism for constructing hierarchical multi-object matching and propagation. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 Identification Mechanism for Multi-object Association ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            782,
+            589,
+            797
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Many recent VOS methods [26, 23, 34] utilized attention mechanisms and achieved promising results. To formulate, we define $Q \\in \\mathbb { R } ^ { H W \\times C }$ , $K \\in \\mathbb { R } ^ { T H W \\times C }$ , and $V \\in \\mathbb { R } ^ { T H W \\times C }$ as the query embedding of the current frame, the key embedding of the memory frames, and the value embedding of the memory frames respectively, where $T$ , $H$ , $W$ , $C$ denote the temporal, height, width, and channel dimensions. The formula of a common attention-based matching and propagation is, ",
+        "bbox": [
+            174,
+            808,
+            825,
+            878
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6272bc964970d7b8a86961da9e5dd2c9153cadfc6c636f1e0a78349cea15fb96.jpg",
+        "text": "$$\nA t t ( Q , K , V ) = C o r r ( Q , K ) V = s o f t m a x ( \\frac { Q K ^ { t r } } { \\sqrt { C } } ) V ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            882,
+            684,
+            916
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where a matching map is calculated by the correlation function Corr, and then the value embedding, $V$ , will be propagated into each location of the current frame. ",
+        "bbox": [
+            173,
+            90,
+            825,
+            119
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In the common single-object propagation [26], the binary mask information in memory frames is embedded into $V$ with an additional memory encoder network and thus can also be propagated to the current frame by using Eq. 2. A convolutional decoder network following the propagated feature will decode the aggregated feature and predict the single-object probability logit of the current frame. ",
+        "bbox": [
+            174,
+            126,
+            825,
+            181
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The main problem of propagating and decoding multi-object mask information in an end-to-end network is how to adapt the network to different target numbers. To overcome this problem, we propose an identification mechanism consisting of identification embedding and decoding based on attention mechanisms. ",
+        "bbox": [
+            174,
+            188,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "First, an Identification Embedding mechanism is proposed to embed the masks of multiple different targets into the same feature space for propagation. As seen in Fig. 2b, we initialize an identity bank, $D \\in \\mathbb { R } ^ { M \\times C }$ , where $M$ identification vectors with $C$ dimensions are stored. For embedding multiple different target masks, each target will be randomly assigned a different identification vector. Assuming $N$ $N < M )$ targets are in the video scenery, the formula of embedding the targets’ one-hot mask, $\\bar { Y } \\in \\{ 0 , 1 \\} ^ { T H W \\times N }$ , into a identification embedding, $E \\in \\mathbb { R } ^ { T H W \\times C }$ , by randomly assigning identification vector from the bank $D$ is, ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a66a306a12822a0c2ed8fd6523a5b5e2acfa39a9f6c2daaba5b4f716afb76ae4.jpg",
+        "text": "$$\nE = I D ( Y , D ) = Y P D ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            413,
+            352,
+            583,
+            369
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $P \\in \\{ 0 , 1 \\} ^ { N \\times M }$ is a random permutation matrix, satisfying that $P ^ { t r } P$ is equal to a $M \\times M$ unit matrix, for randomly selecting $N$ identification embeddings. After the $I D$ assignment, different target has different identification embedding, and thus we can propagate all the target identification information from memory frames to the current frame by attaching the identification embedding $E$ with the attention value $V$ , i.e., ",
+        "bbox": [
+            174,
+            373,
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+            444
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bb90ef7c58e2a6560697512730711309eed38442c22b2f1dedba305fa99dfc82.jpg",
+        "text": "$$\nV ^ { \\prime } = A t t I D ( Q , K , V , Y | D ) = A t t ( Q , K , V + I D ( Y , D ) ) = A t t ( Q , K , V + E ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            227,
+            449,
+            767,
+            467
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $V ^ { \\prime } \\in \\mathbb { R } ^ { H W \\times C }$ aggregates all the multiple targets’ embeddings from the propagation. ",
+        "bbox": [
+            168,
+            472,
+            772,
+            487
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For Identification Decoding, i.e., predicting all the targets’ probabilities from the aggregated feature $V ^ { \\prime }$ , we firstly predict the probability logit for every identity in the bank $D$ by employing a convolutional decoding network $\\hat { F } ^ { \\mathcal { D } }$ , and then select the assigned ones and calculate the probabilities, i.e., ",
+        "bbox": [
+            173,
+            492,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5dc44551b1550deaf423c37ac06ef99d222af885da9d1eeeaf95cae967afb2d0.jpg",
+        "text": "$$\nY ^ { \\prime } = s o f t m a x ( P F ^ { \\mathcal { D } } ( V ^ { \\prime } ) ) = s o f t m a x ( P L ^ { D } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            545,
+            660,
+            564
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $L ^ { D } \\in \\mathbb { R } ^ { H W \\times M }$ is all the $M$ identities’ probability logits, $P$ is the same as the selecting matrix used in the identity assignment (Eq. 3), and $\\begin{array} { r } { Y ^ { \\dot { \\prime } } \\in [ 0 , 1 ] ^ { \\check { H } W \\times N } } \\end{array}$ is the probability prediction of all the $N$ targets. ",
+        "bbox": [
+            176,
+            565,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For training, common multi-class segmentation losses, such as cross-entropy loss, can be used to optimize the multi-object $Y ^ { \\prime }$ regarding the ground-truth labels. The identity bank $D$ is trainable and randomly initialized at the training beginning. To ensure that all the identification vectors have the same opportunity to compete with each other, we randomly reinitialize the identification selecting matrix $P$ in each video sample and each optimization iteration. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 Long Short-Term Transformer for Hierarchical Matching and Propagation ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            699,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Previous methods [26, 34] always utilize only one layer of attention (Eq. 2) to aggregate singleobject information. In our identification-based multi-object pipeline, we found that a single attention layer cannot fully model multi-object association, which naturally should be more complicated than single-object processes. Thus, we consider constructing hierarchical matching and propagation by using a series of attention layers. Recently, transformer blocks [39] have been demonstrated to be stable and promising in constructing hierarchical attention structures in visual tasks [8, 12]. Based on transformer blocks, we carefully design a Long Short-Term Transformer (LSTT) block for multi-object VOS. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Following the common transformer blocks [39, 11], LSTT firstly employs a self-attention layer, which is responsible for learning the association or correlation among the targets within the current frame. Then, LSTT additionally introduces a long-term attention, for aggregating targets’ information from long-term memory frames and a short-term attention, for learning temporal smoothness from nearby short-term frames. The final module is based on a common 2-layer feed-forward MLP with ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "GELU [17] non-linearity in between. Fig. 2c shows the structure of an LSTT block. Notably, all these attention modules are implemented in the form of the multi-head attention [39], i.e., multiple attention modules followed by concatenation and a linear projection. Nevertheless, we only introduce their single-head formulas below for the sake of simplicity. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Long-Term Attention is responsible for aggregating targets’ information from past memory frames, which contains the reference frame and stored predicted frames, to the current frame. Since the time intervals between the current frame and past frames are variable and can be long-term, the temporal smoothness is difficult to guarantee. Thus, the long-term attention employs non-local attention like Eq. 2. Let $X _ { l } ^ { t } \\in \\mathbb { R } ^ { H W \\times \\mathbf { \\breve { C } } }$ denotes the input feature embedding at time $t$ and in block $l$ , where $l \\in \\{ 1 , . . . , L \\}$ is the block index of LSTT, the formula of the long-term attention is, ",
+        "bbox": [
+            173,
+            152,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/298540a67d349cc53f93ca87e2e942cfeca6bf54be7b8a569b53621f45eda71b.jpg",
+        "text": "$$\nA t t L T ( X _ { l } ^ { t } , X _ { l } ^ { \\mathbf { m } } , Y ^ { \\mathbf { m } } ) = A t t I D ( X _ { l } ^ { t } W _ { l } ^ { K } , X _ { l } ^ { \\mathbf { m } } W _ { l } ^ { K } , X _ { l } ^ { \\mathbf { m } } W _ { l } ^ { V } , Y ^ { \\mathbf { m } } | D ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            242,
+            732,
+            262
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $X _ { l } ^ { \\mathbf { m } } = C o n c a t ( X _ { l } ^ { m _ { 1 } } , . . . , X _ { l } ^ { m _ { T } } )$ and $Y ^ { \\mathbf { m } } = { C o n c a t } ( Y ^ { m _ { 1 } } , . . . , Y ^ { m _ { T } } )$ are the input feature embeddings and target masks of memory frames with indices $\\mathbf { m } = \\{ m _ { 1 } , . . . , m _ { T } \\}$ . Besides, $W _ { l } ^ { K } \\in$ $\\mathbb { R } ^ { C \\times C _ { k } }$ and $W _ { l } ^ { V } \\in \\mathbb { R } ^ { C \\times C _ { v } }$ are trainable parameters of the space projections for matching and propagation, respectively. Instead of using different projections for ${ \\bar { X } } _ { l } ^ { t }$ and $X _ { l } ^ { \\mathbf { m } }$ , we found the training of LSTT is more stable with a siamese-like matching, $i . e .$ , matching between the features within the same embedding space $\\mathit { l }$ -th features with the same projection of $\\breve { W } _ { l } ^ { K }$ ). ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Short-Term Attention is employed for aggregating information in a spatial-temporal neighbourhood for each current-frame location. Intuitively, the image changes across several contiguous video frames are always smooth and continuous. Thus, the target matching and propagation in contiguous frames can be restricted in a small spatial-temporal neighborhood, leading to better efficiency than non-local processes. Considering $n$ neighbouring frames with indices $\\mathbf { n } = \\{ t - 1 , . . . , t - n \\}$ are in the spatialtemporal neighbourhood, the features and masks of these frames are $X _ { l } ^ { \\mathbf { n } } = C o n c a t ( X _ { l } ^ { t - 1 } , . . . , X _ { l } ^ { t - n } )$ and $Y ^ { \\mathbf { n } } = C o n c a t ( Y ^ { t - 1 } , . . . , Y ^ { t - n } )$ , and then the formula of the short-term attention at each spatial location $p$ is, ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/350cd618e23980aaf63aedd4474387ae341c106e9beabfee09e7141aea64e2f1.jpg",
+        "text": "$$\nA t t S T ( X _ { l } ^ { t } , X _ { l } ^ { { \\bf n } } , Y ^ { { \\bf n } } | p ) = A t t L T ( X _ { l , p } ^ { t } , X _ { l , N ( p ) } ^ { { \\bf n } } , Y _ { l , N ( p ) } ^ { { \\bf n } } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            472,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $X _ { l , p } ^ { t } \\in \\mathbb { R } ^ { 1 \\times C }$ is the feature of $X _ { l } ^ { t }$ at location $p , { \\mathcal { N } } ( p )$ is a $\\lambda \\times \\lambda$ spatial neighbourhood l,p centered at location $p$ , and thus $X _ { l , N ( p ) } ^ { \\mathbf { n } }$ land $Y _ { l , \\mathcal { N } ( p ) } ^ { \\mathbf { n } }$ are the features and masks of the spatial-temporal neighbourhood, respectively, with a shape of $n \\lambda ^ { 2 } \\times C$ or $n \\lambda ^ { 2 } \\times N$ . ",
+        "bbox": [
+            174,
+            494,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "When extracting features of the first frame $t = 1$ , there is no memory frames or previous frames, and hence we use $\\bar { X } _ { l } ^ { 1 }$ to replace $X _ { l } ^ { \\mathbf { m } }$ and $X _ { l } ^ { \\mathbf { n } }$ . In other words, the long-term attention and the short-term attention are changed into self-attentions without adjusting the network structures and parameters. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 Implementation Details ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            611,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Network Details: For sufficiently validating the effectiveness of our identification mechanism and LSTT, we mainly use light-weight backbone encoder, MobileNet-V2 [33], and decoder, FPN [20] with Group Normalization [45]. The spatial neighborhood size $\\lambda$ is set to 15, and the number of identification vectors, $M$ , is set to 10, which is consistent with the maximum object number in the benchmarks [48, 31]. AOT performs well with PaddlePaddle [1] and PyTorch [28]. More details can be found in the supplementary material. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Architecture Variants: We build several AOT variant networks with different LSTT layer number $L$ or long-term memory size m. The hyper-parameters of these variants are: (1) AOT-Tiny: $L = 1$ , $\\mathbf { m } = \\bar { \\{ 1 \\} }$ ; (2) AOT-Small: $L = 2$ , $\\mathbf { m } = \\{ 1 \\}$ ; (3) AOT-Base: $L = 3$ , $\\mathbf { m } = \\{ 1 \\}$ ; (4) AOT-Large: $L = 3$ , $\\begin{array} { r } { \\dot { \\mathbf { m } } = \\{ 1 , 1 + \\delta , 1 + 2 \\delta , 1 + 3 \\delta , . . . \\} . } \\end{array}$ . In the experiments, we also equip AOT-L with ResNet50 (R50) [16] or Swin-B [22]. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "AOT-S is a small model with only 2 layers of LSTT block. Compared to AOT-S, AOT-T utilizes only 1 layer of LSTT, and AOT- $\\mathbf { B } / \\mathbf { L }$ uses 3 layers. In AOT-T/S/B, only the first frame is considered into long-term memory, which is similar to [41, 50], leading to a smooth efficiency. In AOT-L, the predicted frames are stored into long-term memory per $\\delta$ frames, following the memory reading strategy [26]. We set $\\delta$ to 2/5 for training/testing. ",
+        "bbox": [
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+            806,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Training Details: Following [46, 26, 23, 34], the training stage is divided into two phases: (1) pre-training on sythetic video sequence generated from static image datasets [14, 21, 10, 36, 15] by randomly applying multiple image augmentations [46]. (2) main training on the VOS benchmarks [48, 31] by randomly applying video augmentations [50]. More details are in the supplementary material. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/c44640ca33dd3668c5c530922353c500cd5c8ae513a042dd06c3146e1f53e6fb.jpg",
+        "table_caption": [
+            "Table 1: The quantitative evaluation on multi-object benchmarks, YouTube-VOS [48] and DAVIS 2017 [31]. Y: using YouTube-VOS for training. ∗: using 600p instead of 480p videos in inference. $^ \\ddag$ : timing extrapolated from single-object speed assuming linear scaling in the number of objects. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"7\">(a) YouTube-VOS</td><td colspan=\"5\">(b) DAVIS 2017</td></tr><tr><td></td><td></td><td colspan=\"2\">Seen</td><td colspan=\"3\">Unseen</td><td>Methods</td><td>J&amp;F</td><td>J</td><td>F</td><td>FPS</td></tr><tr><td>Methods</td><td>J&amp;F</td><td>J</td><td>F</td><td>J</td><td>F</td><td>FPS</td><td colspan=\"3\">Validation2017 Split</td><td></td><td></td></tr><tr><td colspan=\"3\">Validation 2018 Split</td><td></td><td></td><td></td><td></td><td>CFBI [50] (Y)</td><td>81.9</td><td>79.3</td><td>84.5</td><td>5.9</td></tr><tr><td>STM[ICCV19] [26]</td><td>79.4</td><td>79.7</td><td>84.2</td><td>72.8</td><td>80.9</td><td>-</td><td>SST[13](Y)</td><td>82.5</td><td>79.9</td><td>85.1</td><td>1</td></tr><tr><td>KMN[ECCV20] [34]</td><td>81.4</td><td>81.4</td><td>85.6</td><td>75.3</td><td>83.3</td><td></td><td>KMN [34]</td><td>76.0</td><td>74.2</td><td>77.8</td><td>4.2t</td></tr><tr><td>CFBI[ECCV20] [50]</td><td>81.4</td><td>81.1</td><td>85.8</td><td>75.3</td><td>83.4</td><td>3.4</td><td>KMN [34] (Y)</td><td>82.8</td><td>80.0</td><td>85.6</td><td>4.2</td></tr><tr><td>LWL[ECCV20] [6]</td><td>81.5</td><td>80.4</td><td>84.9</td><td>76.4</td><td>84.4</td><td>1</td><td>CFBI+ [51] (Y)</td><td>82.9</td><td>80.1</td><td>85.7</td><td>5.6</td></tr><tr><td>SST[CVPR21] [13]</td><td>81.7</td><td>81.2</td><td>1</td><td>76.0</td><td>1</td><td>1</td><td>AOT-T (Y)</td><td>79.9</td><td>77.4</td><td>82.3</td><td>51.4</td></tr><tr><td>CFBI+[TPAMI21] [51]</td><td>82.8</td><td>81.8</td><td>86.6</td><td>77.1</td><td>85.6</td><td>4.0</td><td>AOT-S</td><td>79.2</td><td>76.4</td><td>82.0</td><td>40.0</td></tr><tr><td>AOT-T</td><td>80.2</td><td>80.1</td><td>84.5</td><td>74.0</td><td>82.2</td><td>41.0</td><td>AOT-S (Y)</td><td>81.3</td><td>78.7</td><td>83.9</td><td>40.0</td></tr><tr><td>AOT-S</td><td>82.6</td><td>82.0</td><td>86.7</td><td>76.6</td><td>85.0</td><td>27.1</td><td>AOT-B (Y)</td><td>82.5</td><td>79.7</td><td>85.2</td><td>29.6</td></tr><tr><td>AOT-B</td><td>83.5</td><td>82.6</td><td>87.5</td><td>77.7</td><td>86.0</td><td>20.5</td><td>AOT-L (Y)</td><td>83.8</td><td>81.1</td><td>86.4</td><td>18.7</td></tr><tr><td>AOT-L</td><td>83.8</td><td>82.9</td><td>87.9</td><td>77.7</td><td>86.5</td><td>16.0</td><td>R50-AOT-L (Y)</td><td>84.9</td><td>82.3</td><td>87.5</td><td>18.0</td></tr><tr><td>R50-AOT-L</td><td>84.1</td><td>83.7</td><td>88.5</td><td>78.1</td><td>86.1</td><td>14.9</td><td>SwinB-AOT-L (Y)</td><td>85.4</td><td>82.4</td><td>88.4</td><td>12.1</td></tr><tr><td>SwinB-AOT-L</td><td>84.5</td><td>84.3</td><td>89.3</td><td>77.9</td><td>86.4</td><td>9.3</td><td>Testing 2017 Split</td><td></td><td></td><td></td><td></td></tr><tr><td colspan=\"5\">Validation 2019 Split</td><td></td><td colspan=\"5\">71.4 78.7</td></tr><tr><td>CFBI[ECCV20] [50]</td><td>81.0</td><td>80.6</td><td>85.1</td><td>75.2</td><td>83.0</td><td>3.4</td><td>CFBI [50] (Y) CFBI* [50](Y)</td><td>75.0 76.6</td><td>73.0</td><td>80.1</td><td>5.3 2.9</td></tr><tr><td>SST[CVPR21] [13]</td><td>81.8</td><td>80.9</td><td>1</td><td>76.6</td><td>-</td><td>-</td><td>KMN* [34] (Y)</td><td>77.2</td><td>74.1</td><td>80.3</td><td>1</td></tr><tr><td>CFBI+[TPAMI21] [51]</td><td>82.6</td><td>81.7</td><td>86.2</td><td>77.1</td><td>85.2</td><td>4.0</td><td>CFBI+* [51](Y)</td><td>78.0</td><td>74.4</td><td>81.6</td><td>3.4</td></tr><tr><td>AOT-T</td><td>79.7</td><td>79.6</td><td>83.8</td><td>73.7</td><td>81.8</td><td>41.0</td><td>AOT-T (Y)</td><td>72.0</td><td>68.3</td><td>75.7</td><td>51.4</td></tr><tr><td>AOT-S</td><td>82.2</td><td>81.3</td><td>85.9</td><td>76.6</td><td>84.9</td><td>27.1</td><td>AOT-S (Y)</td><td>73.9</td><td>70.3</td><td>77.5</td><td>40.0</td></tr><tr><td>AOT-B</td><td>83.3</td><td>82.4</td><td>87.1</td><td>77.8</td><td>86.0</td><td>20.5</td><td>AOT-B (Y)</td><td>75.5</td><td>71.6</td><td>79.3</td><td>29.6</td></tr><tr><td>AOT-L</td><td>83.7</td><td>82.8</td><td>87.5</td><td>78.0</td><td>86.7</td><td>16.0</td><td>AOT-L (Y)</td><td>78.3</td><td>74.3</td><td>82.3</td><td>18.7</td></tr><tr><td>R50-AOT-L</td><td>84.1</td><td>83.5</td><td>88.1</td><td>78.4</td><td>86.3</td><td>14.9</td><td>R50-AOT-L (Y)</td><td>79.6</td><td>75.9</td><td>83.3</td><td>18.0</td></tr><tr><td>SwinB-AOT-L</td><td>84.5</td><td>84.0</td><td>88.8</td><td>78.4</td><td>86.7</td><td>9.3</td><td>SwinB-AOT-L (Y)</td><td>81.2</td><td>77.3</td><td>85.1</td><td>12.1</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6 Experimental Results ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We evaluate AOT on popular multi-object benchmarks, YouTube-VOS [48] and DAVIS 2017 [31], and single-object benchmark, DAVIS 2016 [30]. For YouTube-VOS experiments, we train our models on the YouTube-VOS 2019 training split. For DAVIS, we train on the DAVIS-2017 training split. When evaluating YouTube-VOS, we use the default 6FPS videos, and all the videos are restricted to be smaller than $1 . 3 \\times 4 8 0 p$ resolution. As to DAVIS, the default 480p 24FPS videos are used. ",
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+    },
+    {
+        "type": "text",
+        "text": "The evaluation metric is the $\\mathcal { I }$ score, calculated as the average Intersect over Union (IoU) score between the prediction and the ground truth mask, and the $\\mathcal { F }$ score, calculated as an average boundary similarity measure between the boundary of the prediction and the ground truth, and their mean value, denoted as $\\mathcal { I } \\& \\mathcal { F }$ . We evaluate all the results on official evaluation servers or with official tools. ",
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+        "type": "text",
+        "text": "6.1 Compare with the State-of-the-art Methods ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "YouTube-VOS [48] is the latest large-scale benchmark for multi-object video segmentation and is about 37 times larger than DAVIS 2017 (120 videos). Specifically, YouTube-VOS contains 3471 videos in the training split with 65 categories and 474/507 videos in the validation 2018/2019 split with additional 26 unseen categories. The unseen categories do not exist in the training split to evaluate algorithms’ generalization ability. ",
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+        "type": "text",
+        "text": "As shown in Table 1a, AOT variants achieve superior performance on YouTube-VOS compared to the previous state-of-the-art methods. With our identification mechanism, AOT-S ( $8 2 . 6 \\%$ J &F ) ",
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+    {
+        "type": "image",
+        "img_path": "images/d28db8d7b0e6eb707adaa9bbbec03cc11f76dfc9cca42cb535353a8852c56975.jpg",
+        "image_caption": [
+            "Figure 3: Qualitative results. (top) Compared with CFBI [50], AOT performs better when segmenting multiple highly similar objects (carousels and zebras). (bottom) AOT fails to segment some tiny objects (ski poles and watch) since AOT has no specific design for processing rare tiny objects. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "is comparable with $\\mathrm { C F B I + }$ [51] $( 8 2 . 8 \\% )$ while running about $7 \\times$ faster (27.1 vs 4.0FPS). By using more LSTT blocks, AOT-B improves the performance to $8 3 . 5 \\%$ . Moreover, AOT-L further improves both the seen and unseen scores by utilizing the memory reading strategy, and our R50-AOT-L $( 8 4 . 1 \\% / 8 4 . 1 \\% )$ significantly outperforms the previous methods on the validation 2018/2019 split while maintaining an efficient speed (14.9FPS). ",
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+    {
+        "type": "text",
+        "text": "DAVIS 2017 [31] is a multi-object extension of DAVIS 2016. The validation split of DAVIS 2017 consists of 30 videos with 59 objects, and the training split contains 60 videos with 138 objects. Moreover, the testing split contains 30 more challenging videos with 89 objects in total. ",
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+    {
+        "type": "text",
+        "text": "Table 1b shows that our R50-AOT-L $( \\mathbf { Y } )$ surpasses all the competitors on both the DAVIS-2017 validation $( \\mathbf { 8 4 . 9 \\% } )$ and testing $( 7 9 . 6 \\% )$ splits and maintains an efficient speed (18.0FPS). Notably, such a multiobject speed is the same as our single-object speed on DAVIS 2016. For the first time, processing multiple objects can be as efficient as processing a single object over the AOT framework. We also evaluate our method without training with YouTube-VOS, and AOT-S $( 7 9 . 2 \\% )$ performs much better than KMN [34] $( 7 6 . 0 \\% )$ by $+ 3 . 2 \\%$ . ",
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+    {
+        "type": "table",
+        "img_path": "images/348bf4a6b547417548ce288d20925e0e0e53045e09e729a957689f7e8b744780.jpg",
+        "table_caption": [
+            "Table 2: The quantitative evaluation on the single-object DAVIS 2016 [30]. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td>J&amp;F</td><td>J</td><td>F</td><td>FPS</td></tr><tr><td>CFBI+ [51] (Y)</td><td>89.9 90.5</td><td>88.7 89.5</td><td>91.1 91.5</td><td>5.9 8.3</td></tr><tr><td>KMN [34](Y) AOT-T (Y)</td><td>86.8</td><td>86.1</td><td>87.4</td><td>51.4</td></tr><tr><td>AOT-S (Y)</td><td>89.4</td><td></td><td>88.6 90.2</td><td>40.0</td></tr><tr><td>AOT-B (Y)</td><td>89.9</td><td></td><td>88.7 91.1</td><td>29.6</td></tr><tr><td>AOT-L (Y)</td><td>90.4</td><td>89.6 91.1</td><td></td><td>18.7</td></tr><tr><td>R50-AOT-L (Y) SwinB-AOT-L (Y)</td><td>91.1 92.0</td><td></td><td>90.1 92.11 90.7 93.3 12.1</td><td>18.0</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "DAVIS 2016 [30] is a single-object benchmark containing 20 videos in the validation split. Although our AOT aims at improving multi-object video segmentation, we also achieve a new state-of-the-art performance on DAVIS 2016 (R50-AOT-L $( \\mathbf { Y } )$ , $91 . 1 \\%$ ). Under single-object scenarios, the multiobject superiority of AOT is limited, but R50-AOT-L still maintains an about $2 \\times$ efficiency compared to KMN $( 1 8 . 0 \\nu s 8 . 3 \\mathrm { F P S } )$ ). Furthermore, our smaller variant, AOT-B $( 8 9 . 9 \\% )$ , achieves comparable performance with CFBI+ $( 8 9 . 9 \\% )$ while running $5 \\times$ faster (29.6 vs 5.9FPS). ",
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+    {
+        "type": "text",
+        "text": "Apart from the above results, replacing the AOT encoder from commonly used ResNet50 to SwinB can further boost our performance to higher level (Table 1a, 1b, and 2). ",
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+    {
+        "type": "text",
+        "text": "Qualitative results: Fig. 3 visualizes some qualitative results in comparison with CFBI [50], which only associates each object with its relative background. As demonstrated, CFBI is easier to confuse multiple highly similar objects. In contrast, our AOT tracks and segments all the targets accurately by associating all the objects uniformly. However, AOT fails to segment some tiny objects (ski poles and watch) since we do not make special designs for tiny objects. ",
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+    {
+        "type": "text",
+        "text": "6.2 Ablation Study ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we analyze the main components and hyper-parameters of AOT and evaluate their impact on the VOS performance in Table 3. ",
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+    {
+        "type": "text",
+        "text": "Identity number: The number of the identification vectors, $M$ , have to be larger than the object number in videos. Thus, we set $M$ to 10 in default to be consistent with the maximum object number ",
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+    {
+        "type": "text",
+        "text": "(a) Identity number ",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 3: Ablation study. The experiments are based on AOT-S and conducted on the validation 2018 split of YouTube-VOS [48] without pre-training on synthetic videos. Self: the position embedding type used in the self-attention. Rel: use relative positional embedding [35] on the local attention. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>M J&amp;F</td><td>Jseen Junseen</td></tr><tr><td>10 80.3</td><td>80.6</td></tr><tr><td>79.0</td><td>73.7 79.4</td></tr><tr><td></td><td>72.1 70.8</td></tr><tr><td>20 78.3 30 77.2</td><td>79.4 78.5 70.2</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/1794d96dbe224fe96917bf84154b94e453fcbd22d23ef75a1ed6d2f4d0d97af2.jpg",
+        "table_caption": [
+            "(c) Local frame number "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>n J&amp;F</td><td>Jseen</td><td>Junseen</td></tr><tr><td>1 80.3</td><td>80.6</td><td>73.7</td></tr><tr><td>2</td><td>80.0 79.8</td><td>73.7</td></tr><tr><td>3 79.1</td><td>80.0</td><td>72.2</td></tr><tr><td>0 74.3</td><td>74.9</td><td>67.6</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f777949608447e03887eb761751a994c4818c1c70c7ab557adf4003dfd356fe6.jpg",
+        "table_caption": [
+            "(b) Local window size ",
+            "(d) LSTT block number "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>入 J&amp;F</td><td>Jseen</td><td>Junseen</td></tr><tr><td>15 80.3</td><td>80.6</td><td>73.7</td></tr><tr><td>11</td><td>78.8 79.5</td><td>71.9</td></tr><tr><td>7 78.3</td><td>79.3</td><td>70.9</td></tr><tr><td>0 74.3</td><td>74.9</td><td>67.6</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/96f260fcb5115e3e4353ef083e4a885d31a3f345b1ee48930cf82f596f60eeb7.jpg",
+        "table_caption": [
+            "(e) Positional embedding "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\"> Self Rel J&amp;F gseen</td><td></td><td>Junseen</td></tr><tr><td>sine</td><td>√</td><td>80.3</td><td>80.6</td><td>73.7</td></tr><tr><td>none</td><td>√</td><td>80.1</td><td>80.4</td><td>73.5</td></tr><tr><td>sine</td><td>1</td><td>79.7</td><td>80.1</td><td>72.9</td></tr></table>",
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/8fe7954246dbaa89228972c34632a88cb8eff77c57066fca26317596b66150c3.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>L J&amp;F Jseen</td><td></td><td>Junseen</td><td>FPS Param</td></tr><tr><td>2</td><td>80.3 80.6</td><td>73.7</td><td>27.17.0M</td></tr><tr><td>3</td><td>80.9</td><td>81.1 74.0</td><td>20.58.3M</td></tr><tr><td>1</td><td>77.9 78.8</td><td>71.0</td><td>41.0 5.7M</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "in the benchmarks [48, 31]. As seen in Table 3a, $M$ larger than 10 leads to worse performance since (1) no training video contains so many objects; (2) embedding more than 10 objects into the space with only 256 dimensions is difficult. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Local window size: Table 3b shows that larger local window size, $\\lambda$ , results in better performance. Without the local attention, $\\lambda = 0$ , the performance of AOT significantly drops from $8 0 . 3 \\%$ to $7 4 . 3 \\%$ , which demonstrates the necessity of the local attention. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Local frame number: In Table 3c, we also try to employ more previous frames in the local attention, but using only the $t - 1$ frame $( 8 0 . 3 \\% )$ performs better than using 2/3 frames $( 8 0 . 0 \\% / 7 9 . 1 \\% )$ . A possible reason is that the longer the temporal interval, the more intense the motion between frames, so it is easier to introduce more errors in the local matching when using an earlier previous frame. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "LSTT block number: As shown in Table 3d, the AOT performance increases by using more LSTT blocks. Notably, the AOT with only one LSTT block $( 7 7 . 9 \\% )$ reaches a fast real-time speed (41.0FPS) on YouTube-VOS, although the performance is $- 2 . 4 \\%$ worse than AOT-S $( 8 0 . 3 \\% )$ . By adjusting the LSTT block number, we can flexibly balance the accuracy and speed of AOT. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Position embedding: In our default setting, we apply fixed sine spatial positional embedding to the self-attention following [8], and our local attention is equipped with learned relative positional embedding [35]. The ablation study is shown in Table 3e, where removing the sine embedding decreases the performance to $8 0 . 1 \\%$ slightly. In contrast, the relative embedding is more important than the sine embedding. Without the relative embedding, the performance drops to $7 9 . 7 \\%$ , which means the motion relationship between adjacent frames is helpful for local attention. We also tried to apply learned positional embedding to self-attention modules, but no positive effect was observed. ",
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+    {
+        "type": "text",
+        "text": "7 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This paper proposes a novel and efficient approach for video object segmentation by associating objects with transformers and achieves superior performance on three popular benchmarks. A simple yet effective identification mechanism is proposed to associate, match, and decode all the objects uniformly under multi-object scenarios. For the first time, processing multiple objects in VOS can be efficient as processing a single object by using the identification mechanism. In addition, a long short-term transformer is designed for constructing hierarchical object matching and propagation for VOS. The hierarchical structure allows us to flexibly balance AOT between real-time speed and stateof-the-art performance by adjusting the LSTT number. We hope the identification mechanism will help ease the future study of multi-object VOS and related tasks (e.g., video instance segmentation, interactive VOS, and multi-object tracking), and AOT will serve as a solid baseline. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "References ",
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+        "bbox": [
+            174,
+            89,
+            267,
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+    },
+    {
+        "type": "text",
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